In
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the standard
deviation is a measure of the amount of variation or
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
of a set of values.
A low standard deviation indicates that the values tend to be close to the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
(also called the
expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case
Greek letter
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
σ (sigma), for the population standard deviation, or the
Latin letter ''
s'', for the sample standard deviation.
The standard deviation of a
random variable,
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of ...
,
statistical population,
data set, or
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
is the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of its
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
. It is
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
ically simpler, though in practice less
robust, than the
average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data.
The standard deviation of a population or sample and the
standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the
margin of error of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population.
In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered
"statistically significant", a safeguard against spurious conclusion that is really due to random sampling error.
When only a
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of ...
of data from a population is available, the term ''standard deviation of the sample'' or ''sample standard deviation'' can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the ''population standard deviation'' (the standard deviation of the entire population).
Basic examples
Population standard deviation of grades of eight students
Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of the
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (that is, a
statistical population) are the following eight values:
:
These eight data points have the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
(average) of 5:
:
First, calculate the deviations of each data point from the mean, and
square the result of each:
:
The
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
is the mean of these values:
:
and the ''population'' standard deviation is equal to the square root of the variance:
:
This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by instead of in the denominator of the last formula, and the result is
In that case, the result of the original formula would be called the ''sample'' standard deviation and denoted by ''s'' instead of
Dividing by ''n'' − 1 rather than by ''n'' gives an unbiased estimate of the variance of the larger parent population. This is known as ''
Bessel's correction''. Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by ''n'' would underestimate the variability.
Standard deviation of average height for adult men
If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the
average height for adult men in the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
is about 70 inches, with a standard deviation of around 3 inches. This means that most men (about 68%, assuming a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches tall. If the standard deviation were 20 inches, then men would have much more variable heights, with a typical range of about 50–90 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is
normal or bell-shaped (see the
68–95–99.7 rule
In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within
an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, t ...
, or the ''empirical rule,'' for more information).
Definition of population values
Let ''μ'' be the
expected value (the average) of
random variable ''X'' with density ''f''(''x''):
The standard deviation ''σ'' of ''X'' is defined as
which can be shown to equal
Using words, the standard deviation is the square root of the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of ''X''.
The standard deviation of a probability distribution is the same as that of a random variable having that distribution.
Not all random variables have a standard deviation. If the distribution has
fat tails
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are somet ...
going out to infinity, the standard deviation might not exist, because the integral might not converge. The
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The
Pareto distribution with parameter
has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The
Cauchy distribution has neither a mean nor a standard deviation.
Discrete random variable
In the case where ''X'' takes random values from a finite data set ''x''
1, ''x''
2, …, ''x
N'', with each value having the same probability, the standard deviation is
or, by using
summation notation,
If, instead of having equal probabilities, the values have different probabilities, let ''x''
1 have probability ''p''
1, ''x''
2 have probability ''p''
2, …, ''x''
''N'' have probability ''p''
''N''. In this case, the standard deviation will be
Continuous random variable
The standard deviation of a
continuous real-valued random variable ''X'' with
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''p''(''x'') is
and where the integrals are
definite integrals taken for ''x'' ranging over the set of possible values of the random variable ''X''.
In the case of a
parametric family of distributions, the standard deviation can be expressed in terms of the parameters. For example, in the case of 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 normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
with parameters ''μ'' and ''σ''
2, the standard deviation is
Estimation
One can find the standard deviation of an entire population in cases (such as
standardized testing) where every member of a population is sampled. In cases where that cannot be done, the standard deviation ''σ'' is estimated by examining a random sample taken from the population and computing a
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an
estimator, and the estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by ''s'' (possibly with modifiers).
Unlike in the case of estimating the population mean, for which the
sample mean is a simple estimator with many desirable properties (
unbiased,
efficient, maximum likelihood), there is no single estimator for the standard deviation with all these properties, and
unbiased estimation of standard deviation is a very technically involved problem. Most often, the standard deviation is estimated using the ''
corrected sample standard deviation'' (using ''N'' − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using ''N'') yields lower mean squared error, while using ''N'' − 1.5 (for the normal distribution) almost completely eliminates bias.
Uncorrected sample standard deviation
The formula for the ''population'' standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by ''s''
''N'', is known as the ''uncorrected sample standard deviation'', or sometimes the ''standard deviation of the sample'' (considered as the entire population), and is defined as follows:
where
are the observed values of the sample items, and
is the mean value of these observations, while the denominator ''N'' stands for the size of the sample: this is the square root of the sample variance, which is the average of the
squared deviations about the sample mean.
This is a
consistent estimator (it converges in probability to the population value as the number of samples goes to infinity), and is the
maximum-likelihood estimate when the population is normally distributed. However, this is a
biased estimator, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/''N'', and thus is most significant for small or moderate sample sizes; for
the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller
mean squared error than the corrected sample standard deviation.
Corrected sample standard deviation
If the ''biased
sample variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
'' (the second
central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is
:
Here taking the square root introduces further downward bias, by
Jensen's inequality, due to the square root's being a
concave function. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question.
An unbiased estimator for the ''variance'' is given by applying
Bessel's correction, using ''N'' − 1 instead of ''N'' to yield the ''unbiased sample variance,'' denoted ''s''
2:
:
This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement. ''N'' − 1 corresponds to the number of
degrees of freedom in the vector of deviations from the mean,
Taking square roots reintroduces bias (because the square root is a nonlinear function which does not
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with the expectation, i.e. often
), yielding the ''corrected sample standard deviation,'' denoted by ''s:''
:
As explained above, while ''s''
2 is an unbiased estimator for the population variance, ''s'' is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples (''N'' less than 10). As sample size increases, the amount of bias decreases. We obtain more information and the difference between
and
becomes smaller.
Unbiased sample standard deviation
For
unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, ''s'' is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by ''s''/''c''
4, where the correction factor (which depends on ''N'') is given in terms of the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
, and equals:
:
This arises because the sampling distribution of the sample standard deviation follows a (scaled)
chi distribution, and the correction factor is the mean of the chi distribution.
An approximation can be given by replacing ''N'' − 1 with ''N'' − 1.5, yielding:
:
The error in this approximation decays quadratically (as 1/''N''
2), and it is suited for all but the smallest samples or highest precision: for ''N'' = 3 the bias is equal to 1.3%, and for ''N'' = 9 the bias is already less than 0.1%.
A more accurate approximation is to replace
above with
.
For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:
:
where ''γ''
2 denotes the population
excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.
Confidence interval of a sampled standard deviation
The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the
confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
or CI.
To show how a larger sample will make the confidence interval narrower, consider the following examples:
A small population of ''N'' = 2 has only 1 degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD;
the factors here are as follows:
:
where
is the ''p''-th quantile of the chi-square distribution with ''k'' degrees of freedom, and
is the confidence level. This is equivalent to the following:
:
With ''k'' = 1,
and
. The reciprocals of the square roots of these two numbers give us the factors 0.45 and 31.9 given above.
A larger population of ''N'' = 10 has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69 × SD to 1.83 × SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a sample population N=100, this is down to 0.88 × SD to 1.16 × SD. To be more certain that the sampled SD is close to the actual SD we need to sample a large number of points.
These same formulae can be used to obtain confidence intervals on the variance of residuals from a
least squares fit under standard normal theory, where ''k'' is now the number of
degrees of freedom for error.
Bounds on standard deviation
For a set of ''N'' > 4 data spanning a range of values ''R'', an upper bound on the standard deviation ''s'' is given by ''s = 0.6R''.
An estimate of the standard deviation for ''N'' > 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values ''R'' represents four standard deviations so that ''s ≈ R/4''. This so-called range rule is useful in
sample size
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populati ...
estimation, as the range of possible values is easier to estimate than the standard deviation. Other divisors ''K(N)'' of the range such that ''s ≈ R/K(N)'' are available for other values of ''N'' and for non-normal distributions.
Identities and mathematical properties
The standard deviation is invariant under changes in
location, and scales directly with the
scale of the random variable. Thus, for a constant ''c'' and random variables ''X'' and ''Y'':
:
The standard deviation of the sum of two random variables can be related to their individual standard deviations and the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
between them:
:
where
and
stand for variance and
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
, respectively.
The calculation of the sum of squared deviations can be related to
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
s calculated directly from the data. In the following formula, the letter E is interpreted to mean expected value, i.e., mean.
:
The sample standard deviation can be computed as:
:
For a finite population with equal probabilities at all points, we have
:
which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.
See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.
Interpretation and application
A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.
For example, each of the three populations , and has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. These standard deviations have the same units as the data points themselves. If, for instance, the data set represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters.
Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated
measurements gives the
precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified. See
prediction interval
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
.
While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the
mean absolute deviation, which might be considered a more direct measure of average distance, compared to the
root mean square distance inherent in the standard deviation.
Application examples
The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).
Experiment, industrial and hypothesis testing
Standard deviation is often used to compare real-world data against a model to test the model.
For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (99.9% or more). If it falls outside the range then the production process may need to be corrected. Statistical tests such as these are particularly important when the testing is relatively expensive. For example, if the product needs to be opened and drained and weighed, or if the product was otherwise used up by the test.
In experimental science, a theoretical model of reality is used.
Particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
conventionally uses a standard of "5 sigma" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to assert that a particle consistent with the
Higgs boson had been discovered in two independent experiments at
CERN
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
, also leading to the declaration of the
first observation of gravitational waves.
Weather
As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.
Finance
In finance, standard deviation is often used as a measure of the
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (known as
mean-variance optimization). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the risk premium. In other words, investors should expect a higher return on an investment when that investment carries a higher level of risk or uncertainty. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns.
For example, assume an investor had to choose between two stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20
percentage points (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return). Stock B is likely to fall short of the initial investment (but also to exceed the initial investment) more often than Stock A under the same circumstances, and is estimated to return only two percent more on average. In this example, Stock A is expected to earn about 10 percent, plus or minus 20 pp (a range of 30 percent to −10 percent), about two-thirds of the future year returns. When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70 percent to −50 percent, which includes outcomes for three standard deviations from the average return (about 99.7 percent of probable returns).
Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average gives the overall variance of the return of the asset. The larger the variance, the greater risk the security carries. Finding the square root of this variance will give the standard deviation of the investment tool in question.
Population standard deviation is used to set the width of
Bollinger Bands, a
technical analysis tool. For example, the upper Bollinger Band is given as
The most commonly used value for ''n'' is 2; there is about a five percent chance of going outside, assuming a normal distribution of returns.
Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.
Geometric interpretation
To gain some geometric insights and clarification, we will start with a population of three values, ''x''
1, ''x''
2, ''x''
3. This defines a point ''P'' = (''x''
1, ''x''
2, ''x''
3) in R
3. Consider the line ''L'' = . This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and ''P'' would lie on ''L''. So it is not unreasonable to assume that the standard deviation is related to the ''distance'' of ''P'' to ''L''. That is indeed the case. To move orthogonally from ''L'' to the point ''P'', one begins at the point:
:
whose coordinates are the mean of the values we started out with.
is on
therefore
for some
.
The line
is to be orthogonal to the vector from
to
. Therefore:
:
A little algebra shows that the distance between ''P'' and ''M'' (which is the same as the
orthogonal distance between ''P'' and the line ''L'')
is equal to the standard deviation of the vector (''x''
1, ''x''
2, ''x''
3), multiplied by the square root of the number of dimensions of the vector (3 in this case).
Chebyshev's inequality
An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.
Rules for normally distributed data
The
central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of
:
where ''μ'' is the
expected value of the random variables, ''σ'' equals their distribution's standard deviation divided by ''n''
1/2, and ''n'' is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the
normalizing constant.
If a data distribution is approximately normal, then the proportion of data values within ''z'' standard deviations of the mean is defined by:
:
where
is the
error function. The proportion that is less than or equal to a number, ''x'', is given by the
cumulative distribution function:
: