HOME

TheInfoList

Click Here for Items Related To - Gaussian profile

OR:

Gaussian profile on: [Wikipedia] [Google] [Amazon]

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, a normal distribution or Gaussian distribution is a type of

continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

for a

real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

. The general form of its

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

f(x) = \frac e^\,.

The parameter is the

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

or expectation of the distribution (and also its

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

and mode), while the parameter

\sigma^2

is the

variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...

. The

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in

and are often used in the

natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...

and

social science Social science (often rendered in the plural as the social sciences) is one of the branches of science, devoted to the study of societies and the relationships among members within those societies. The term was formerly used to refer to the ...

s to represent real-valued

s whose distributions are not known. Their importance is partly due to the

central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...

. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as

measurement error Observational error (or measurement error) is the difference between a measured value of a quantity and its unknown true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. Such errors are inherent in the measurement pr ...

s, often have distributions that are nearly normal. Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as

propagation of uncertainty In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of ex ...

and

least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed. A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the

Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

, Student's ''t'', and logistic distributions). (For other names, see ''

Naming Naming is assigning a name to something. Naming may refer to: * Naming (parliamentary procedure), a procedure in certain parliamentary bodies * Naming ceremony, an event at which an infant is named * Product naming, the discipline of deciding wha ...

''.) The univariate probability distribution is generalized for vectors in the

multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...

and for matrices in the

matrix normal distribution In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density ...

Definitions

Standard normal distribution

The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when

\mu=0

and

\sigma^2 =1

, and it is described by this

(or density):

\varphi(z) = \frac\,.

The variable has a mean of 0 and a variance and standard deviation of 1. The density

\varphi(z)

has its peak

\frac

z=0

and

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...

s at

z=+1

and . Although the density above is most commonly known as the ''standard normal,'' a few authors have used that term to describe other versions of the normal distribution.

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

, for example, once defined the standard normal as

\varphi(z) = \frac,

which has a variance of , and

Stephen Stigler Stephen Mack Stigler (born August 10, 1941) is the Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. He has authored several books on the history of statistics; he is the son of ...

once defined the standard normal as

\varphi(z) = e^,

which has a simple functional form and a variance of

\sigma^2 = \frac .

General normal distribution

Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor (the standard deviation) and then translated by (the mean value):

f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac \sigma \right)\,.

The probability density must be scaled by

1/\sigma

so that the

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

is still 1. If is a

standard normal deviate Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object t ...

, then

X=\sigma Z + \mu

will have a normal distribution with expected value and standard deviation . This is equivalent to saying that the standard normal distribution can be scaled/stretched by a factor of and shifted by to yield a different normal distribution, called . Conversely, if is a normal deviate with parameters and

\sigma^2

, then this distribution can be re-scaled and shifted via the formula

Z=(X-\mu)/\sigma

to convert it to the standard normal distribution. This variate is also called the standardized form of .

Notation

The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter (

phi Phi ( ; uppercase Φ, lowercase φ or ϕ; ''pheî'' ; Modern Greek: ''fi'' ) is the twenty-first letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th to 4th century BC), it represented an aspirated voiceless bilabial plos ...

). The alternative form of the Greek letter phi, , is also used quite often. The normal distribution is often referred to as

N(\mu,\sigma^2)

or . Thus when a random variable is normally distributed with mean and standard deviation , one may write

X \sim \mathcal(\mu,\sigma^2).

Alternative parameterizations

Some authors advocate using the precision as the parameter defining the width of the distribution, instead of the standard deviation or the variance . The precision is normally defined as the reciprocal of the variance, . The formula for the distribution then becomes

f(x) = \sqrt e^.

This choice is claimed to have advantages in numerical computations when is very close to zero, and simplifies formulas in some contexts, such as in the

Bayesian inference Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian infer ...

of variables with

. Alternatively, the reciprocal of the standard deviation

\tau'=1/\sigma

might be defined as the ''precision'', in which case the expression of the normal distribution becomes

f(x) = \frac e^.

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the

quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...

s of the distribution. Normal distributions form an

exponential family In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...

with

natural parameter In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...

\textstyle\theta_1=\frac

and

\textstyle\theta_2=\frac

, and natural statistics ''x'' and ''x''². The dual expectation parameters for normal distribution are and .

Cumulative distribution function

The

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

(CDF) of the standard normal distribution, usually denoted with the capital Greek letter , is the integral

\Phi(x) = \frac 1  \int_^x e^ \, dt\,.

Error function

The related

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

\operatorname(x)

gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range . That is:

\operatorname(x) = \frac 1  \int_^x e^ \, dt = \frac 2  \int_0^x e^ \, dt\,.

These integrals cannot be expressed in terms of elementary functions, and are often said to be

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

s. However, many numerical approximations are known; see

below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...

for more. The two functions are closely related, namely

\Phi(x) = \frac \left + \operatorname\left( \frac x  \right) \right,.

For a generic normal distribution with density , mean and variance

\sigma^2

, the cumulative distribution function is

F(x) = \Phi = \frac \left + \operatorname\left(\frac\right)\right,.

The complement of the standard normal cumulative distribution function,

Q(x) = 1 - \Phi(x)

, is often called the Q-function, especially in engineering texts. It gives the probability that the value of a standard normal random variable will exceed : . Other definitions of the -function, all of which are simple transformations of , are also used occasionally. The

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

of the standard normal cumulative distribution function has 2-fold

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...

around the point (0,1/2); that is, . Its

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...

(indefinite integral) can be expressed as follows:

\int \Phi(x)\, dx = x\Phi(x) + \varphi(x) + C.

The cumulative distribution function of the standard normal distribution can be expanded by

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...

into a series:

\Phi(x)=\frac + \frac\cdot e^ \left + \frac + \frac + \cdots + \frac + \cdots\right,.

where

!!

denotes the

double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...

. An

asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...

of the cumulative distribution function for large ''x'' can also be derived using integration by parts. For more, see . A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation:

\Phi(x) \approx \frac+\frac \sum_^n \frac\,.

Recursive computation with Taylor series expansion

The recursive nature of the

e^

family of derivatives may be used to easily construct a rapidly converging

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

expansion using recursive entries about any point of known value of the distribution,

\Phi(x_0)

\Phi(x) = \sum_^\infty \frac(x-x_0)^n\,,

where:

\begin
\Phi^(x_0) &= \frac\int_^e^\,dt \\
\Phi^(x_0) &= \frace^ \\
\Phi^(x_0) &= -\left(x_0\Phi^(x_0)+(n-2)\Phi^(x_0)\right), & n \geq 2\,.
\end

Using the Taylor series and Newton's method for the inverse function

An application for the above Taylor series expansion is to use

Newton's method In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...

to reverse the computation. That is, if we have a value for the

\Phi(x)

, but do not know the x needed to obtain the

\Phi(x)

, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of

\Phi(x)

, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution. To solve, select a known approximate solution,

x_0

, to the desired .

x_0

may be a value from a distribution table, or an intelligent estimate followed by a computation of

\Phi(x_0)

using any desired means to compute. Use this value of

x_0

and the Taylor series expansion above to minimize computations. Repeat the following process until the difference between the computed

\Phi(x_)

and the desired , which we will call

\Phi(\text)

, is below a chosen acceptably small error, such as 10⁻⁵, 10⁻¹⁵, etc.:

x_ = x_n - \frac\,,

where :

\Phi(x,x_0,\Phi(x_0))

is the

\Phi(x)

from a Taylor series solution using

x_0

and

\Phi(x_0)

\Phi'(x_n)=\frace^\,.

When the repeated computations converge to an error below the chosen acceptably small value, ''x'' will be the value needed to obtain a

\Phi(x)

of the desired value, .

Standard deviation and coverage

About 68% of values drawn from a normal distribution are within one standard deviation ''σ'' from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68–95–99.7 (empirical) rule, or the ''3-sigma rule''. More precisely, the probability that a normal deviate lies in the range between

\mu-n\sigma

and

\mu+n\sigma

is given by

F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname \left(\frac\right).

To 12 significant digits, the values for

n=1,2,\ldots , 6

are: For large , one can use the approximation

1 - p \approx \frac

Quantile function

The

quantile function In probability and statistics, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distr ...

of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse

\Phi^(p) = \sqrt2\operatorname^(2p - 1), \quad p\in(0,1).

For a normal random variable with mean and variance

\sigma^2

, the quantile function is

F^(p) = \mu + \sigma\Phi^(p)
= \mu + \sigma\sqrt 2 \operatorname^(2p - 1), \quad p\in(0,1).

The

\Phi^(p)

of the standard normal distribution is commonly denoted as . These values are used in

hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...

, construction of confidence intervals and

Q–Q plot In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...

s. A normal random variable will exceed

\mu + z_p\sigma

with probability

1-p

, and will lie outside the interval

\mu \pm z_p\sigma

with probability . In particular, the quantile

z_

is 1.96; therefore a normal random variable will lie outside the interval

\mu \pm 1.96\sigma

in only 5% of cases. The following table gives the quantile

z_p

such that will lie in the range

\mu \pm z_p\sigma

with a specified probability . These values are useful to determine

tolerance interval A tolerance interval (TI) is a statistical interval within which, with some confidence level, a specified sampling (statistics), sampled proportion of a population falls. "More specifically, a tolerance interval provides limits within which at l ...

for sample averages and other statistical

estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...

s with normal (or

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...

ally normal) distributions. The following table shows

\sqrt 2 \operatorname^(p)=\Phi^\left(\frac\right)

, not

\Phi^(p)

as defined above. For small , the quantile function has the useful

\Phi^(p)=-\sqrt+\mathcal(1).

Properties

The normal distribution is the only distribution whose

cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...

s beyond the first two (i.e., other than the mean and

) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance. Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184 The normal distribution is a subclass of the

elliptical distribution In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...

s. The normal distribution is

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the

weight In science and engineering, the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition. Some sta ...

of a person or the price of a share. Such variables may be better described by other distributions, such as the

log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...

or the

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial scien ...

. The value of the normal density is practically zero when the value lies more than a few

s away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of

outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...

s—values that lie many standard deviations away from the mean—and least squares and other

statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...

methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied. The Gaussian distribution belongs to the family of

stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...

s which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the

Cauchy distribution The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...

and the

Lévy distribution In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...

Symmetries and derivatives

The normal distribution with density

f(x)

(mean and variance

\sigma^2 > 0

) has the following properties: * It is symmetric around the point

x=\mu,

which is at the same time the mode, the

and the

of the distribution. * It is

unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal p ...

: its first

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

is positive for

x<\mu,

negative for

x>\mu,

and zero only at

x=\mu.

* The area bounded by the curve and the -axis is unity (i.e. equal to one). * Its first derivative is

f'(x)=-\frac f(x).

* Its second derivative is

f''(x) =  \frac f(x).

* Its density has two

s (where the second derivative of is zero and changes sign), located one standard deviation away from the mean, namely at

x=\mu-\sigma

and

x=\mu+\sigma.

* Its density is log-concave. * Its density is infinitely

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

, indeed supersmooth of order 2. Furthermore, the density of the standard normal distribution (i.e.

\mu=0

and

\sigma=1

) also has the following properties: * Its first derivative is

\varphi'(x)=-x\varphi(x).

* Its second derivative is

\varphi''(x)=(x^2-1)\varphi(x)

* More generally, its th derivative is

\varphi^(x) = (-1)^n\operatorname_n(x)\varphi(x),

where

\operatorname_n(x)

is the th (probabilist)

Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...

. * The probability that a normally distributed variable with known and

\sigma^2

is in a particular set, can be calculated by using the fact that the fraction

Z = (X-\mu)/\sigma

has a standard normal distribution.

Moments

The plain and absolute moments of a variable are the expected values of

X^p

and

, X, ^p

, respectively. If the expected value of is zero, these parameters are called ''central moments;'' otherwise, these parameters are called ''non-central moments.'' Usually we are interested only in moments with integer order . If has a normal distribution, the non-central moments exist and are finite for any whose real part is greater than −1. For any non-negative integer , the plain central moments are:

= \begin 0 & \textp\text \\ \sigma^p (p-1)!! & \textp\text \end

Here

n!!

denotes the

, that is, the product of all numbers from to 1 that have the same parity as

n.

The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer

p,

\begin
   \operatorname\left confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...

_1F_1

and

U.

\begin
   \operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
   \operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .


The expectation of  conditioned on the event that  lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where  and  respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density  of  is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .

Fourier transform and characteristic function

The

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

of a normal density with mean and variance

\sigma^2

\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,

where is the

imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

. If the mean

\mu=0

, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the

frequency domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...

, with mean 0 and variance . In particular, the standard normal distribution is an

eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

of the Fourier transform. In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points ...

\varphi_X(t)

of that variable, which is defined as the

expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...

e^

, as a function of the real variable (the

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:

\varphi_X(t) = \hat f(-t)\,.

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of

e^

, as a function of the real parameter . For a normal distribution with density , mean and variance

\sigma^2

, the moment generating function exists and is equal to

= \hat f(it) = e^ e^\,.

For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value . The

cumulant generating function In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...

is the logarithm of the moment generating function, namely

g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two

s are nonzero, namely the mean and the variance . Some authors prefer to instead work with the

and .

Stein operator and class

Within

Stein's method Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972, to obtain ...

the Stein operator and class of a random variable

X \sim \mathcal(\mu, \sigma^2)

are

\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)

and

\mathcal

the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when

\sigma^2

tends to zero, the probability density

f(x)

eventually tends to zero at any

x\ne \mu

, but grows without limit if

x = \mu

, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

when . However, one can define the normal distribution with zero variance as a

generalized function In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...

; specifically, as a

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

translated by the mean , that is

f(x)=\delta(x-\mu).

Its cumulative distribution function is then the

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...

translated by the mean , namely

F(x) =
\begin
  0 & \textx < \mu \\
  1 & \textx \geq \mu\,.
\end

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution

N(\mu,\sigma^2)

is the one with maximum entropy. To see this, let be a

continuous random variable In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...

with probability density . The entropy of is defined as

H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,

where

f(x)\log f(x)

is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using

variational calculus The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

. A function with three

Lagrange multipliers In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...

is defined:

L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.

At maximum entropy, a small variation

\delta f(x)

about

f(x)

will produce a variation

\delta L

about which is equal to 0:

0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that

f(x)=\frace^\,.

The entropy of a normal distribution

X \sim N(\mu,\sigma^2)

is equal to

H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,

which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where

X_1,\ldots ,X_n

are

independent and identically distributed Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...

random variables with the same arbitrary distribution, zero mean, and variance

\sigma^2

and is their mean scaled by

\sqrt

Z = \sqrt\left(\frac\sum_^n X_i\right)

Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance . The theorem can be extended to variables

(X_i)

that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions. Many

test statistic Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...

s, scores, and

s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions. The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: * The

binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...

B(n,p)

is approximately normal with mean

np

and variance

np(1-p)

for large and for not too close to 0 or 1. * The

Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...

with parameter is approximately normal with mean and variance , for large values of . * The

chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...

\chi^2(k)

is approximately normal with mean and variance

2k

, for large . * The

Student's t-distribution In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...

t(\nu)

is approximately normal with mean 0 and variance 1 when is large. Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution. A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions. This theorem can also be used to justify modeling the sum of many uniform noise sources as

Gaussian noise Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

. See

AWGN Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: * ''Additive'' because it is added to any nois ...

Operations and functions of normal variables

Probabilities of functions of normal vectors

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance

\sigma^2

, then *

aX+b

, for any real numbers and , is also normally distributed, with mean

a\mu+b

and variance

a^2\sigma^2

. That is, the family of normal distributions is closed under

linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

. * The exponential of is distributed log-normally:

e^X \sim \ln(N(\mu, \sigma^2))

. * The standard

sigmoid Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include: * Sigmoid function, a mathematical function * Sigmoid colon, part of the l ...

of is logit-normally distributed:

\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )

. * The absolute value of has

folded normal distribution The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...

. If

\mu = 0

this is known as the

half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...

. * The absolute value of normalized residuals,

, X - \mu,  / \sigma

, has

chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...

with one degree of freedom:

, X - \mu,  / \sigma \sim \chi_1

. * The square of

X/\sigma

has the

noncentral chi-squared distribution In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...

with one degree of freedom:

X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)

. If

\mu = 0

, the distribution is called simply chi-squared. * The log-likelihood of a normal variable is simply the log of its

\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).

Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable. * The distribution of the variable restricted to an interval

, b The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

/math> is called the

truncated normal distribution In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...

. *

(X - \mu)^

has a

with location 0 and scale

\sigma^

= Operations on two independent normal variables

= * If

X_1

and

X_2

are two

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...

normal random variables, with means

\mu_1

\mu_2

and variances

\sigma_1^2

\sigma_2^2

, then their sum

X_1 + X_2

will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$\begin
\operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
\operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .

The expectation of conditioned on the event that lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$\begin
\operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
\operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .

The expectation of conditioned on the event that lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$&= \sigma^p \cdot 2^ \frac \, _1F_1. \end$

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$\begin
\operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
\operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .

The expectation of conditioned on the event that lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$\begin
\operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
\operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .

The expectation of conditioned on the event that lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$\begin
\operatorname\left^p\right &= \sigma^p\cdot ^p \, U, \\
\operatorname\left Hermite polynomials#"Negative variance", generalized Hermite polynomials .

The expectation of conditioned on the event that lies in an interval,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X - \mu, ^p\right&= \sigma^p (p-1)!! \cdot \begin
\sqrt & \textp\text \\
1 & \textp\text
\end \\
&= \sigma^p \cdot \frac.
\end
The last formula is valid also for any non-integer $p>-1.$ When the mean $\mu \ne 0,$ the plain and absolute moments can be expressed in terms of confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...
s $_1F_1$ and $U.$

$&= \sigma^p \cdot 2^ \frac \, _1F_1. \end$

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions>X, ^p \right&= \sigma^p \cdot 2^ \frac \, _1F_1.
\end

These expressions remain valid even if is not an integer. See also Hermite polynomials#"Negative variance", generalized Hermite polynomials.

The expectation of conditioned on the event that lies in an interval $,b /math> is given by \operatorname\left \mid a = \mu - \sigma^2\frac\,, where and respectively are the density and the cumulative distribution function of . For b=\infty this is known as the inverse Mills ratio . Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have \sigma^2 instead of .$
Fourier transform and characteristic function

The Fourier transform

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a normal density with mean and variance $\sigma^2$ is

$\hat f(t) = \int_^\infty f(x)e^ \, dx = e^ e^\,,$

where is the imaginary unit

The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
. If the mean $\mu=0$ , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, with mean 0 and variance . In particular, the standard normal distribution is an eigenfunction

In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable is closely connected to the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X(t)$ of that variable, which is defined as the expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of $e^$ , as a function of the real variable (the frequency

Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable . The relation between both is:
$\varphi_X(t) = \hat f(-t)\,.$

Moment- and cumulant-generating functions

The moment generating function of a real random variable is the expected value of $e^$ , as a function of the real parameter . For a normal distribution with density , mean and variance $\sigma^2$ , the moment generating function exists and is equal to

$= \hat f(it) = e^ e^\,.$
For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
is the logarithm of the moment generating function, namely

$g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.$

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s are nonzero, namely the mean and the variance .

Some authors prefer to instead work with the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
and .

Stein operator and class

Within Stein's method

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972,
to obtain ...
the Stein operator and class of a random variable $X \sim \mathcal(\mu, \sigma^2)$ are $\mathcalf(x) = \sigma^2 f'(x) - (x-\mu)f(x)$ and $\mathcal$ the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit when $\sigma^2$ tends to zero, the probability density $f(x)$ eventually tends to zero at any $x\ne \mu$ , but grows without limit if $x = \mu$ , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function

Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
when .

However, one can define the normal distribution with zero variance as a generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
; specifically, as a Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
translated by the mean , that is $f(x)=\delta(x-\mu).$
Its cumulative distribution function is then the Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
translated by the mean , namely
$F(x) =
\begin
0 & \textx < \mu \\
1 & \textx \geq \mu\,.
\end$

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance , the normal distribution $N(\mu,\sigma^2)$ is the one with maximum entropy. To see this, let be a continuous random variable

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...
with probability density . The entropy of is defined as
$H(X) = - \int_^\infty f(x)\ln f(x)\, dx\,,$

where $f(x)\log f(x)$ is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. A function with three Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
is defined:

$L=-\int_^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_^\infty f(x)(x-\mu)^2\,dx\right)\,.$

At maximum entropy, a small variation $\delta f(x)$ about $f(x)$ will produce a variation $\delta L$ about which is equal to 0:

$0=\delta L=\int_^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.$

Since this must hold for any small , the factor multiplying must be zero, and solving for yields:

$f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.$

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and are chosen so that
$f(x)=\frace^\,.$
The entropy of a normal distribution $X \sim N(\mu,\sigma^2)$ is equal to
$H(X)=\tfrac(1+\ln 2\sigma^2\pi)\,,$
which is independent of the mean .

Other properties

Related distributions

Central limit theorem

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $X_1,\ldots ,X_n$ are independent and identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
random variables with the same arbitrary distribution, zero mean, and variance $\sigma^2$ and is their
mean scaled by $\sqrt$
$Z = \sqrt\left(\frac\sum_^n X_i\right)$
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

The theorem can be extended to variables $(X_i)$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistic

Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
s, scores, and estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
$B(n,p)$ is approximately normal with mean $np$ and variance $np(1-p)$ for large and for not too close to 0 or 1.
* The Poisson distribution

In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with parameter is approximately normal with mean and variance , for large values of .
* The chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
$\chi^2(k)$ is approximately normal with mean and variance $2k$ , for large .
* The Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
$t(\nu)$ is approximately normal with mean 0 and variance 1 when is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. See AWGN

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.

Operations and functions of normal variables

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing
Matlab code
. In the following sections we look at some special cases.

Operations on a single normal variable

If is distributed normally with mean and variance $\sigma^2$ , then
* $aX+b$ , for any real numbers and , is also normally distributed, with mean $a\mu+b$ and variance $a^2\sigma^2$ . That is, the family of normal distributions is closed under linear transformations

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.
* The exponential of is distributed log-normally: $e^X \sim \ln(N(\mu, \sigma^2))$ .
* The standard sigmoid
Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include:

* Sigmoid function, a mathematical function
* Sigmoid colon, part of the l ...
of is logit-normally distributed: $\sigma(X) \sim P( \mathcal(\mu,\,\sigma^2) )$ .
* The absolute value of has folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
: . If $\mu = 0$ this is known as the half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
.
* The absolute value of normalized residuals, $, X - \mu, / \sigma$ , has chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. E ...
with one degree of freedom: $, X - \mu, / \sigma \sim \chi_1$ .
* The square of $X/\sigma$ has the noncentral chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...
with one degree of freedom: $X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)$ . If $\mu = 0$ , the distribution is called simply chi-squared.
* The log-likelihood of a normal variable is simply the log of its probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: $\ln p(x)= -\frac \left(\frac \right)^2 -\ln \left(\sigma \sqrt \right).$ Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
* The distribution of the variable restricted to an interval , b

The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is called the truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ....
* $(X - \mu)^$ has a Lévy distribution

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
with location 0 and scale $\sigma^$ .

= Operations on two independent normal variables
=
* If $X_1$ and $X_2$ are two independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
normal random variables, with means $\mu_1$ , $\mu_2$ and variances $\sigma_1^2$ , $\sigma_2^2$ , then their sum $X_1 + X_2$ will also be normally distributed,^{roof/sup> with mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$ .
* In particular, if and are independent normal deviates with zero mean and variance $\sigma^2$ , then $X + Y$ and $X - Y$ are also independent and normally distributed, with zero mean and variance $2\sigma^2$ . This is a special case of the polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product t ...
.
* If $X_1$ , $X_2$ are two independent normal deviates with mean and variance $\sigma^2$ , and , are arbitrary real numbers, then the variable $X_3 = \frac + \mu$ is also normally distributed with mean and variance $\sigma^2$ . It follows that the normal distribution is stable

A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles

There are many different types of stables in use tod ...
(with exponent $\alpha=2$ ).
* If $X_k \sim \mathcal N(m_k, \sigma_k^2)$ , $k \in \$ are normal distributions, then their normalized geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
$\frac X_0^ X_1^$ is a normal distribution $\mathcal N(m_, \sigma_^2)$ with $m_ = \frac$ and $\sigma_^2 = \frac$ .

= Operations on two independent standard normal variables
=
If $X_1$ and $X_2$ are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: $X_1 \pm X_2 \sim \mathcal(0, 2)$ .
* Their product $Z = X_1 X_2$ follows the product distribution
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables ''X'' and ''Y'', the distribution of ...
with density function $f_Z(z) = \pi^ K_0(, z, )$ where $K_0$ is the modified Bessel function of the second kind

Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
. This distribution is symmetric around zero, unbounded at $z = 0$ , and has the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
$\phi_Z(t) = (1 + t^2)^$ .
* Their ratio follows the standard Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...
: $X_1/ X_2 \sim \operatorname(0, 1)$ .
* Their Euclidean norm $\sqrt$ has the Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distributi ...
.

Operations on multiple independent normal variables

* Any linear combination

In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of independent normal deviates is a normal deviate.
* If $X_1, X_2, \ldots, X_n$ are independent standard normal random variables, then the sum of their squares has the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom $X_1^2 + \cdots + X_n^2 \sim \chi_n^2.$
* If $X_1, X_2, \ldots, X_n$ are independent normally distributed random variables with means and variances $\sigma^2$ , then their sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
is independent from the sample standard deviation

In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with $n-1$ degrees of freedom: $t = \frac = \frac \sim t_.$
* If $X_1, X_2, \ldots, X_n$ , $Y_1, Y_2, \ldots, Y_m$ are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution

In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...
with degrees of freedom: $F = \frac \sim F_.$

Operations on multiple correlated normal variables

* A quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
of a normal vector, i.e. a quadratic function $q = \sum x_i^2 + \sum x_j + c$ of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...
results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean and variance $\sigma^2$ is the distribution of the sum of independent normal deviates, each with mean $\frac$ and variance $\frac$ . This property is called infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
.

Conversely, if $X_1$ and $X_2$ are independent random variables and their sum $X_1+X_2$ has a normal distribution, then both $X_1$ and $X_2$ must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.

The Kac–Bernstein theorem

The Kac–Bernstein theorem states that if $X$ and are independent and $X + Y$ and $X - Y$ are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.

More generally, if $X_1, \ldots, X_n$ are independent random variables, then two distinct linear combinations $\sum$ and $\sum$ will be independent if and only if all $X_k$ are normal and $\sum$ , where $\sigma_k^2$ denotes the variance of $X_k$ .

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...
describes the Gaussian law in the -dimensional Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. In the simplified two and three dimensional case, the joint distribution f ...
s. As such, its iso-density loci in the case are ellipse

In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and in the case of arbitrary are ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
s.
* Rectified Gaussian distribution

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...
a rectified version of normal distribution with all the negative elements reset to 0.
* Complex normal distribution

In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''locatio ...
deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the ' matrix , and the ' matrix .
* Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
describes the case of normally distributed matrices.
* Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
es are the normally distributed stochastic process

In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es. These can be viewed as elements of some infinite-dimensional Hilbert space

In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product

In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' . Several Gaussian processes became popular enough to have their own names:
** Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
;
** Brownian bridge; and
** Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
* Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
is an abstract mathematical construction that represents a q-analogue of the normal distribution.
* the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution

In mathematical physics and probability

Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the ...
above.
* The Kaniadakis -Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable has a two-piece normal distribution if it has a distribution

$f_X( x ) = \begin
N( \mu, \sigma_1^2 ),& \text x \le \mu \\
N( \mu, \sigma_2^2 ),& \text x \ge \mu
\end$

where is the mean and and are the variances of the distribution to the left and right of the mean respectively.

The mean , variance , and third central moment of this distribution have been determined

$\end$

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.
History
The Pearson syste ...
— a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The generalized normal distribution

The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...
, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate

Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
them. That is, having a sample $(x_1, \ldots, x_n)$ from a normal $\mathcal(\mu, \sigma^2)$ population we would like to learn the approximate values of parameters and $\sigma^2$ . The standard approach to this problem is the maximum likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
method, which requires maximization of the '' log-likelihood function'':
$\ln\mathcal(\mu,\sigma^2)
= \sum_^n \ln f(x_i\mid\mu,\sigma^2)
= -\frac\ln(2\pi) - \frac\ln\sigma^2 - \frac\sum_^n (x_i-\mu)^2.$
Taking derivatives with respect to and $\sigma^2$ and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
$\hat = \overline \equiv \frac\sum_^n x_i, \qquad
\hat^2 = \frac \sum_^n (x_i - \overline)^2.$

Then $\ln\mathcal(\hat,\hat^2)$ is as follows:

$\ln\mathcal(\hat,\hat^2)
= (-n/2) ln(2 \pi \hat^2)+1 /math>$ Sample mean

Estimator $\textstyle\hat\mu$ is called the ''sample mean

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.

The sample mean is the average value (or me ...
'', since it is the arithmetic mean of all observations. The statistic $\textstyle\overline$ is complete

Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
and sufficient for , and therefore by the Lehmann–Scheffé theorem, $\textstyle\hat\mu$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:
$\hat\mu \sim \mathcal(\mu,\sigma^2/n).$
The variance of this estimator is equal to the ''μμ''-element of the inverse Fisher information matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance ...
$\textstyle\mathcal^$ . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error

The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, it is the standard deviati ...
of $\textstyle\hat\mu$ is proportional to $\textstyle1/\sqrt$ , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulation

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be det ...
s.

From the standpoint of the asymptotic theory, $\textstyle\hat\mu$ is consistent

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, that is, it converges in probability
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
to as $n\rightarrow\infty$ . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
$\sqrt(\hat\mu-\mu) \,\xrightarrow\, \mathcal(0,\sigma^2).$

Sample variance

The estimator $\textstyle\hat\sigma^2$ is called the ''sample variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...
'', since it is the variance of the sample ( $(x_1, \ldots, x_n)$ ). In practice, another estimator is often used instead of the $\textstyle\hat\sigma^2$ . This other estimator is denoted $s^2$ , and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root is called the ''sample standard deviation''. The estimator $s^2$ differs from $\textstyle\hat\sigma^2$ by having instead of ''n'' in the denominator (the so-called Bessel's correction

In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in ...
):
$s^2 = \frac \hat\sigma^2 = \frac \sum_^n (x_i - \overline)^2.$
The difference between $s^2$ and $\textstyle\hat\sigma^2$ becomes negligibly small for large ''n''s. In finite samples however, the motivation behind the use of $s^2$ is that it is an unbiased estimator

In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of the underlying parameter $\sigma^2$ , whereas $\textstyle\hat\sigma^2$ is biased. Also, by the Lehmann–Scheffé theorem the estimator $s^2$ is uniformly minimum variance unbiased (UMVU In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For pra ...
), which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator $\textstyle\hat\sigma^2$ is better than the $s^2$ in terms of the mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(MSE) criterion. In finite samples both $s^2$ and $\textstyle\hat\sigma^2$ have scaled chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with degrees of freedom:
$s^2 \sim \frac \cdot \chi^2_, \qquad
\hat\sigma^2 \sim \frac \cdot \chi^2_.$
The first of these expressions shows that the variance of $s^2$ is equal to $2\sigma^4/(n-1)$ , which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix $\textstyle\mathcal^$ , which is $2\sigma^4/n$ . Thus, $s^2$ is not an efficient estimator for $\sigma^2$ , and moreover, since $s^2$ is UMVU, we can conclude that the finite-sample efficient estimator for $\sigma^2$ does not exist.

Applying the asymptotic theory, both estimators $s^2$ and $\textstyle\hat\sigma^2$ are consistent, that is they converge in probability to $\sigma^2$ as the sample size $n\rightarrow\infty$ . The two estimators are also both asymptotically normal:
$\sqrt(\hat\sigma^2 - \sigma^2) \simeq
\sqrt(s^2-\sigma^2) \,\xrightarrow\, \mathcal(0,2\sigma^4).$
In particular, both estimators are asymptotically efficient for $\sigma^2$ .

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean $\textstyle\hat\mu$ and the sample variance ''s''² are independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
, which means there can be no gain in considering their joint distribution

A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\textstyle\hat\mu$ and ''s'' can be employed to construct the so-called ''t-statistic'':
$t = \frac = \frac \sim t_$
This quantity ''t'' has the Student's t-distribution

In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
with degrees of freedom, and it is an ancillary statistic In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no i ...
(independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the confidence interval for ''μ''; similarly, inverting the ''χ''² distribution of the statistic ''s''² will give us the confidence interval for ''σ''²:
$\mu \in \left \hat\mu - t_ \frac,\,
\hat\mu + t_ \frac \right /math> \sigma^2 \in \left \fracs^2,\,
\fracs^2\right /math>
where ''t$ _k,p'' and are the ''p''th quantile

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s of the ''t''- and ''χ''²-distributions respectively. These confidence intervals are of the '' confidence level'' , meaning that the true values ''μ'' and ''σ''² fall outside of these intervals with probability (or significance level
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
) ''α''. In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''².

Approximate formulas can be derived from the asymptotic distributions of $\textstyle\hat\mu$ and ''s''²:
$\mu \in
\left \hat\mu - \fracs,\,
\hat\mu + \fracs \right /math> \sigma^2 \in
\left s^2 - \sqrt\frac s^2 ,\,
s^2 + \sqrt\frac s^2 \right /math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''$ _''α''/2 do not depend on ''n''. In particular, the most popular value of , results in .
Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis

The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
''H''₀ is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''², versus the alternative ''H_a'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* Q–Q plot

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a List of graphical methods, graphical method for comparing two probability distributions by plotting their ''quantiles'' against each other. A point on the plot ...
, also known as normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(''p_k''), ''x''_(''k'')), where plotting points ''p_k'' are equal to ''p_k'' = (''k'' − ''α'')/(''n'' + 1 − 2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* P–P plot

In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model. It works b ...
– similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''_(''k'')), ''p_k''), where $\textstyle z_ = (x_-\hat\mu)/\hat\sigma$ . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
Goodness-of-fit tests:

''Moment-based tests'':
* D'Agostino's K-squared test
* Jarque–Bera test
* Shapiro–Wilk test

The Shapiro–Wilk test is a Normality test, test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory
The Shapiro–Wilk test tests the null hypothesis that a statistical sample, sample ''x''1, ..., ''x'n'' ...
: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* Anderson–Darling test
* Lilliefors test (an adaptation of the Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional ...
)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and multivariate cases need to be considered.
* Either conjugate or improper prior distribution

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
s may be placed on the unknown variables.
* An additional set of cases occurs in Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficient

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s. The resulting analysis is similar to the basic cases of independent identically distributed
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
article.

Sum of two quadratics

= Scalar form
=
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac\right)^2 + \frac(y-z)^2$

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
. Note the following about the complex constant factors attached to some of the terms:
# The factor $\frac$ has the form of a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of ''y'' and ''z''.
# $\frac = \frac = (a^ + b^)^.$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
, so it is not surprising that $\frac$ is one-half the harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments.

The harmonic mean ...
of ''a'' and ''b''.

= Vector form
=
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ''k'', and A and B are symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, invertible matrices

In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
of size $k\times k$ , then

$\begin
& (\mathbf-\mathbf)'\mathbf(\mathbf-\mathbf) + (\mathbf-\mathbf)' \mathbf(\mathbf-\mathbf) \\
= & (\mathbf - \mathbf)'(\mathbf+\mathbf)(\mathbf - \mathbf) + (\mathbf - \mathbf)'(\mathbf^ + \mathbf^)^(\mathbf - \mathbf)
\end$

where

$\mathbf = (\mathbf + \mathbf)^(\mathbf\mathbf + \mathbf \mathbf)$

The form x′ A x is called a quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2

is a quadratic form in the variables and . The coefficients usually belong t ...
and is a scalar:
$\mathbf'\mathbf\mathbf = \sum_a_ x_i x_j$
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since $x_i x_j = x_j x_i$ , only the sum $a_ + a_$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric

Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Furthermore, if A is symmetric, then the form $\mathbf'\mathbf\mathbf = \mathbf'\mathbf\mathbf.$

Sum of differences from the mean

Another useful formula is as follows:
$\sum_^n (x_i-\mu)^2 = \sum_^n (x_i-\bar)^2 + n(\bar -\mu)^2$
where $\bar = \frac \sum_^n x_i.$

With known variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if $x \sim \mathcal(\mu, 1/\tau)$ and $\mu \sim \mathcal(\mu_0, 1/\tau_0),$ we proceed as follows.

First, the likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
is (using the formula above for the sum of differences from the mean):

$\end$

Then, we proceed as follows:

$\sqrt \exp\left(-\frac\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac\left(\tau\left(\sum_^n(x_i-\bar)^2 + n(\bar -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac \left(n\tau(\bar-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2 + \frac(\bar - \mu_0)^2\right) \\ &\propto \exp\left(-\frac(n\tau + \tau_0)\left(\mu - \dfrac\right)^2\right) \end$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving ''μ''. The result is the kernel of a normal distribution, with mean $\frac$ and precision $n\tau + \tau_0$ , i.e.

$p(\mu\mid\mathbf) \sim \mathcal\left(\frac, \frac\right)$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$\bar &= \frac\sum_^n x_i \end$

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''²) and mean of values $\bar$ , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a weighted average

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis

Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian

Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elde ...
of conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$\bar &= \frac\sum_^n x_i \end$

With known mean

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with known mean μ, the conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
of the variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
has an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac~\frac \propto \frac$

The likelihood function

A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
from above, written in terms of the variance, is:

$\end$

where

$S = \sum_^n (x_i-\mu)^2.$

Then:

$\end$

The above is also a scaled inverse chi-squared distribution where

$\begin
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\mu)^2
\end$

or equivalently

$\begin
\nu_0' &= \nu_0 + n \\
' &= \frac
\end$

Reparameterizing in terms of an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
, the result is:

$\begin
\alpha' &= \alpha + \frac \\
\beta' &= \beta + \frac
\end$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size ''n'' where each individual point ''x'' follows $x \sim \mathcal(\mu, \sigma^2)$ with unknown mean μ and unknown variance

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
σ², a combined (multivariate) conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
is placed over the mean and variance, consisting of a normal-inverse-gamma distribution.
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistic
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It ...
s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate prior

In Bayesian probability theory, if, given a likelihood function
p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posteri ...
s used (an inverse gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$\begin
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end$

The update equations can be derived, and look as follows:

$\begin
\bar &= \frac 1 n \sum_^n x_i \\
\mu_0' &= \frac \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'' &= \nu_0 \sigma_0^2 + \sum_^n (x_i-\bar)^2 + \frac(\mu_0 - \bar)^2
\end$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for $\nu_0''$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
; and
# Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:
* The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).
* The ground state

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
in position space of the quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
.
* The position of a particle that experiences diffusion

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the diffusion equation

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
$\frac f(x,t) = \frac \frac f(x,t)$ . If the initial location is given by a certain density function $g(x)$ , then the density at time ''t'' is the convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of ''g'' and the normal probability density function.

Approximate normality

''Approximately'' normal distributions occur in many situations, as explained by the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...
and decomposable distributions are involved, such as
** Binomial random variables, associated with binary response variables;
** Poisson random variables, associated with rare events;
* Thermal radiation

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.
* In biology

Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
, the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
(after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Compo ...
, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crash

A stock market crash is a sudden dramatic decline of stock prices across a major cross-section of a stock market, resulting in a significant loss of paper wealth. Crashes are driven by panic selling and underlying economic factors. They often fol ...
es. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb

Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist. His work concerns problems of randomness, ...
in his works.
* Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
* In standardized testing
A standardized test is a test that is administered and scored in a consistent or standard manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predetermine ...
, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test

An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. Originally, IQ was a score obtained by dividing a person's mental age score, obtained by administering ...
) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT

The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...
's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

* Many scores are derived from the normal distribution, including percentile rank
In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is

: PR = \frac \times 100,

where ''CF''—the cumulative fr ...
s (percentiles or quantiles), normal curve equivalents, stanine

Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two.

Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric leg ...
s, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests

Student's ''t''-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's ''t''-dis ...
and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
* In hydrology

Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis

John P. A. Ioannidis ( ; , ; born August 21, 1965) is a Greek-American physician-scientist, writer and Stanford University professor who has made contributions to evidence-based medicine, epidemiology, and clinical research. Ioannidis studies sc ...
argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.

Computational methods

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a can be generated as , where ''Z'' is standard normal. All these algorithms rely on the availability of a random number generator

Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular ou ...
''U'' capable of producing uniform

A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
random variates.
* The most straightforward method is based on the probability integral transform
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
property: if ''U'' is distributed uniformly on (0,1), then Φ⁻¹(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.
* An easy-to-program approximate approach that relies on the central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The Box–Muller method uses two independent random numbers ''U'' and ''V'' distributed uniformly on (0,1). Then the two random variables ''X'' and ''Y'' $X = \sqrt \, \cos(2 \pi V) , \qquad
Y = \sqrt \, \sin(2 \pi V) .$ will both have the standard normal distribution, and will be independent
Independent or Independents may refer to:

Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
. This formulation arises because for a bivariate normal random vector (''X'', ''Y'') the squared norm will have the chi-squared distribution

In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities $X = U\sqrt, \qquad Y = V\sqrt$ are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = (''V'' − 0.5)/''U'';
** Optional: if ''X''² ≤ 5 − 4''e''^1/4''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''² ≥ 4''e''^−1.35/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''² ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
* The ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number ge ...
is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ''ideal approximation''; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation into the connection between the fast Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Ever ...
is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
, Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ...
and continued fractions

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...
. Different approximations are used depending on the desired level of accuracy.
* give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error (algorith
26.2.17
: $\Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac,$ where ''ϕ''(''x'') is the standard normal probability density function, and ''b''₀ = 0.2316419, ''b''₁ = 0.319381530, ''b''₂ = −0.356563782, ''b''₃ = 1.781477937, ''b''₄ = −1.821255978, ''b''₅ = 1.330274429.
* lists some dozens of approximations – by means of rational functions, with or without exponentials – for the function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
* suggested a simple algorithm based on the Taylor series expansion $\Phi(x) = \frac12 + \varphi(x)\left( x + \frac 3 + \frac + \frac + \frac + \cdots \right)$ for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).
* The GNU Scientific Library

The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomial

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with tr ...
s.
* proposes the following approximation of $1-\Phi$ with a maximum relative error less than $2^$ $\left(\approx 1.1 \times 10^\right)$ in absolute value: for $x \ge 0$ $\begin
1-\Phi\left(x\right) & = \left(\frac\right)
\left(\frac \right) \\
& \left(\frac\right)
\left(\frac\right) \\
& \left( \frac\right)
\left( \frac\right) e^
\end$ and for $x<0$ ,
$1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting , the simplest approximation for the quantile function is:
$\qquad p\ge 1/2$

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for , corresponding to ). For replace ''p'' by and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
$z=-0.4115\left\, \qquad p\ge 1/2$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
$\text \\ L(z) & \approx \begin 0.4115\left\, & p < 1/2, \\ \\ 0.4115 \dfrac p, & p\ge 1/2. \end \end$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small ''relative'' error on the whole domain for the cumulative distribution function and the quantile function $\Phi^$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his '' The Doctrine of Chances'' the study of the coefficients in the binomial expansion of . De Moivre proved that the middle term in this expansion has the approximate magnitude of $2^n/\sqrt$ , and that "If ''m'' or ''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is $-\frac$ ." Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

In 1823 Gauss

Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
published his monograph "''Theoria combinationis observationum erroribus minimis obnoxiae''" where among other things he introduces several important statistical concepts, such as the method of least squares

The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
, the method of maximum likelihood, and the ''normal distribution''. Gauss used ''M'', , to denote the measurements of some unknown quantity ''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:
$\varphi\mathit = \frac h \, e^,$
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
method.

Although Gauss was the first to suggest the normal distribution law, Laplace

Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
, which emphasized the theoretical importance of the normal distribution.

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.

In the middle of the 19th century Maxwell
Maxwell may refer to:

People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of N ...
demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena: The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x'' + ''dx'' is
$\operatorname \frac\; e^ \, dx$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual. However, by the end of the 19th century some authors had started using the name ''normal distribution'', where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances." Around the turn of the 20th century Pearson popularized the term ''normal'' as a designation for this distribution.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
$df = \frac e^ \, dx.$

The term ''standard normal distribution'', which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A. M. Mood (1950) ''Introduction to the Theory of Statistics''. explicitly defines the ''standard normal distribution'
(p. 112)

See also

* Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
* Bhattacharyya distance – method used to separate mixtures of normal distributions
* Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely ...
– on the occurrence of the normal distribution in number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

* Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve ...

* Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, ...
– convolution

In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, which uses the normal distribution as a kernel
* Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...

* Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
with the pdf on $(0, \infty)$ is given as $f(x)= \frac$ , where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox–Wright Psi function.
* Normally distributed and uncorrelated does not imply independent

Normality is a behavior that can be normal for an individual (intrapersonal normality) when it is consistent with the most common behavior for that person. Normal is also used to describe individual behavior that conforms to the most common beha ...

* Ratio normal distribution
* Reciprocal normal distribution
* Standard normal table

In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic ...

* Stein's lemma
* Sub-Gaussian distribution
* Sum of normally distributed random variables
* Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and th ...
– The normal distribution is a member of the family of Tweedie exponential dispersion models.
* Wrapped normal distribution – the Normal distribution applied to a circular domain
* Z-test

A ''Z''-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. ''Z''-test tests the mean of a distribution. For each statistical significance, signi ...
– using the normal distribution

Notes

References

Citations

Sources

*
* In particular, the entries fo
"bell-shaped and bell curve"
an

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* Translated by Stephen M. Stigler in ''Statistical Science'' 1 (3), 1986: .
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*

Normal distribution calculator

{{Authority control

Continuous distributions
Conjugate prior distributions
Exponential family distributions
Stable distributions
Location-scale family probability distributions

HOME

Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne

Consider donating to Wikimedia

As an Amazon Associate I earn from qualifying purchases}}}}}}}}}}}}}}}}