Noncentral distributions are families of

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 phenomeno ...

s that are related to other "central" families of distributions by means of a noncentrality parameter. Whereas the central distribution describes how a

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

is distributed when the difference tested is

null Null may refer to: Science, technology, and mathematics Computing *Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value *Null character, the zero-valued ASCII character, also designated by , often used ...

, noncentral distributions describe the distribution of a test statistic when the null is false (so the

alternative hypothesis In statistical hypothesis testing, the alternative hypothesis is one of the proposed proposition in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting ...

is true). This leads to their use in calculating

statistical power In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...

. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family. For example, the Student's ''t''-distribution is the central family of distributions for the noncentral ''t''-distribution family. Noncentrality parameters are used in the following distributions: :*

Noncentral t-distribution The noncentral ''t''-distribution generalizes Student's ''t''-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic ''t'' is distributed when the difference tested is null, t ...

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 a ...

:* Noncentral chi-distribution :*

Noncentral F-distribution In probability theory and statistics, the noncentral ''F''-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) ''F''-distribution. It describes the distribution of the quotient (''X''/''n' ...

Noncentral beta distribution In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a noncentral generalization of the (central) beta distribution. The noncentral beta distribution (Type I) is the distributio ...

In general, noncentrality parameters occur in distributions that are transformations of 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 i ...

. The "central" versions are derived from normal distributions that have a

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 '' ari ...

of zero; the noncentral versions generalize to arbitrary means. For example, the standard (central)

chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...

is the distribution of a sum of squared independent

standard normal 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 is ...

distributions, i.e., normal distributions with mean 0,

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 number ...

1. 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 a ...

generalizes this to normal distributions with arbitrary mean and variance. Each of these distributions has a single noncentrality parameter. However, there are extended versions of these distributions which have two noncentrality parameters: the doubly noncentral beta distribution, the doubly noncentral F distribution and the doubly noncentral ''t'' distribution.Johnson, N.L., Kotz, S., Balakrishnan N. (1995). ''Continuous univariate distributions, Volume 2 (2nd Edition)''. Wiley. {{ISBN, 0-471-58494-0 These types of distributions occur for distributions that are defined as the quotient of two independent distributions. When both source distributions are central (either with a zero mean or a zero noncentrality parameter, depending on the type of distribution), the result is a central distribution. When one is noncentral, a (singly) noncentral distribution results, while if both are noncentral, the result is a doubly noncentral distribution. As an example, a ''t''-distribution is defined (ignoring constant values) as the quotient of a normal distribution and the square root of an independent

. Extending this definition to encompass a normal distribution with arbitrary mean produces a

noncentral t-distribution The noncentral ''t''-distribution generalizes Student's ''t''-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic ''t'' is distributed when the difference tested is null, t ...

, while further extending it to allow a

in the denominator while produces a doubly noncentral t-distribution. There are some "noncentral distributions" that are not usually formulated in terms of a "noncentrality parameter": see noncentral hypergeometric distributions, for example. The noncentrality parameter of the ''t''-distribution may be negative or positive while the noncentral parameters of the other three distributions must be greater than zero.

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Online calculator for critical values, cumulative probabilities, and critical noncentral parameters
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