Chisquared distribution
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Template:Probability distribution
In probability theory and statistics, the chisquared distribution (also chisquare or χ²distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. A special case of the gamma distribution, it is one of the most widely used probability distributions in inferential statistics, e.g., in hypothesis testing or in construction of confidence intervals.^{[1]}^{[2]}^{[3]}^{[4]} When it is being distinguished from the more general noncentral chisquared distribution, this distribution is sometimes called the central chisquared distribution.
The chisquared distribution is used in the common chisquared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, like Friedman's analysis of variance by ranks.
Contents
History and name
This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 18756,Template:Sfn^{[5]} where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmertsche ("Helmertian") or "Helmert distribution".
The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chisquared test, published in Template:Harv, with computed table of values published in Template:Harv, collected in Template:Harv. The name "chisquared" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing ½χ² for what would appear in modern notation as ½x^{T}Σ^{−1}x (Σ being the covariance matrix).^{[6]} The idea of a family of "chisquared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.Template:Sfn
Definition
If Z_{1}, ..., Z_{k} are independent, standard normal random variables, then the sum of their squares,
is distributed according to the chisquared distribution with k degrees of freedom. This is usually denoted as
The chisquared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i.e. the number of Z_{i}’s)
Characteristics
Further properties of the chisquared distribution can be found in the box at the upper right corner of this article.
Probability density function
The probability density function (pdf) of the chisquared distribution is
where Γ(k/2) denotes the Gamma function, which has closedform values for integer k.
For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chisquared distribution.
Differential equation
The pdf of the chisquared distribution is a solution to the following differential equation:
Cumulative distribution function
Its cumulative distribution function is:
where γ(s,t) is the lower incomplete Gamma function and P(s,t) is the regularized Gamma function.
In a special case of k = 2 this function has a simple form:
and the form is not much more complicated for other small even k.
Tables of the chisquared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.
Letting , Chernoff bounds on the lower and upper tails of the CDF may be obtained.^{[7]} For the cases when (which include all of the cases when this CDF is less than half):
The tail bound for the cases when , similarly, is
For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chisquared distribution.
Additivity
It follows from the definition of the chisquared distribution that the sum of independent chisquared variables is also chisquared distributed. Specifically, if {X_{i}}_{i=1}^{n} are independent chisquared variables with {k_{i}}_{i=1}^{n} degrees of freedom, respectively, then Y = X_{1} + ⋯ + X_{n} is chisquared distributed with k_{1} + ⋯ + k_{n} degrees of freedom.
Sample mean
The sample mean of n i.i.d. chisquared variables of degree k is distributed according to a gamma distribution with shape α and scale θ parameters:
Asymptotically, given that for a scale parameter going to infinity, a Gamma distribution converges towards a Normal distribution with expectation and variance , the sample mean converges towards:
Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chisquared variable of degree the expectation is , and its variance (and hence the variance of the sample mean being ).
Entropy
The differential entropy is given by
where ψ(x) is the Digamma function.
The chisquared distribution is the maximum entropy probability distribution for a random variate X for which and are fixed. Since the chisquared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the Log moment of Gamma. For derivation from more basic principles, see the derivation in moment generating function of the sufficient statistic.
Noncentral moments
The moments about zero of a chisquared distribution with k degrees of freedom are given by^{[8]}^{[9]}
Cumulants
The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
Asymptotic properties
By the central limit theorem, because the chisquared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k > 50 the distribution is sufficiently close to a normal distribution for the difference to be ignored.^{[10]} Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of tends to a standard normal distribution. However, convergence is slow as the skewness is and the excess kurtosis is 12/k.
 The sampling distribution of ln(χ^{2}) converges to normality much faster than the sampling distribution of χ^{2},^{[11]} as the logarithm removes much of the asymmetry.^{[12]} Other functions of the chisquared distribution converge more rapidly to a normal distribution. Some examples are:
 If X ~ χ²(k) then is approximately normally distributed with mean and unit variance (result credited to R. A. Fisher).
 If X ~ χ²(k) then is approximately normally distributed with mean and variance ^{[13]} This is known as the Wilson–Hilferty transformation.
Relation to other distributions
 As , (normal distribution)
 (Noncentral chisquared distribution with noncentrality parameter )
 (The squared norm of k standard normally distributed variables is a chisquared distribution with k degrees of freedom)
 If and , then . (gamma distribution)
 If then (chi distribution)
 If , then is an exponential distribution. (See Gamma distribution for more.)
 If (Rayleigh distribution) then
 If (Maxwell distribution) then
 If then (Inversechisquared distribution)
 The chisquared distribution is a special case of type 3 Pearson distribution
 If and are independent then (beta distribution)
 If (uniform distribution) then
 is a transformation of Laplace distribution
 chisquared distribution is a transformation of Pareto distribution
 Student's tdistribution is a transformation of chisquared distribution
 Student's tdistribution can be obtained from chisquared distribution and normal distribution
 Noncentral beta distribution can be obtained as a transformation of chisquared distribution and Noncentral chisquared distribution
 Noncentral tdistribution can be obtained from normal distribution and chisquared distribution
A chisquared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.
If Y is a kdimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Y−μ)^{T}C^{−1}(Y−μ) is chisquared distributed with k degrees of freedom.
The sum of squares of statistically independent unitvariance Gaussian variables which do not have mean zero yields a generalization of the chisquared distribution called the noncentral chisquared distribution.
If Y is a vector of k i.i.d. standard normal random variables and A is a k×k symmetric, idempotent matrix with rank k−n then the quadratic form Y^{T}AY is chisquared distributed with k−n degrees of freedom.
The chisquared distribution is also naturally related to other distributions arising from the Gaussian. In particular,
 Y is Fdistributed, Y ~ F(k_{1},k_{2}) if where X_{1} ~ χ²(k_{1}) and X_{2} ~ χ²(k_{2}) are statistically independent.
 If X is chisquared distributed, then is chi distributed.
 If X_{1} ~ χ^{2}_{k1} and X_{2} ~ χ^{2}_{k2} are statistically independent, then X_{1} + X_{2} ~ χ^{2}_{k1+k2}. If X_{1} and X_{2} are not independent, then X_{1} + X_{2} is not chisquared distributed.
Generalizations
The chisquared distribution is obtained as the sum of the squares of k independent, zeromean, unitvariance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.
Linear combination
If are chi square random variables and , then a closed expression for the distribution of is not known. It may be, however, calculated using the property of characteristic functions of the chisquared random variable.^{[14]}
Chisquared distributions
Noncentral chisquared distribution
{{#invoke:mainmain}} The noncentral chisquared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
Generalized chisquared distribution
{{#invoke:mainmain}} The generalized chisquared distribution is obtained from the quadratic form z′Az where z is a zeromean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.
The chisquared distribution X ~ χ²(k) is a special case of the gamma distribution, in that X ~ Γ(k/2, 1/2) using the rate parameterization of the gamma distribution (or X ~ Γ(k/2, 2) using the scale parameterization of the gamma distribution) where k is an integer.
Because the exponential distribution is also a special case of the Gamma distribution, we also have that if X ~ χ²(2), then X ~ Exp(1/2) is an exponential distribution.
The Erlang distribution is also a special case of the Gamma distribution and thus we also have that if X ~ χ²(k) with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.
Applications
The chisquared distribution has numerous applications in inferential statistics, for instance in chisquared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student’s tdistribution. It enters all analysis of variance problems via its role in the Fdistribution, which is the distribution of the ratio of two independent chisquared random variables, each divided by their respective degrees of freedom.
Following are some of the most common situations in which the chisquared distribution arises from a Gaussiandistributed sample.
 if X_{1}, ..., X_{n} are i.i.d. N(μ, σ^{2}) random variables, then where .
 The box below shows some statistics based on X_{i} ∼ Normal(μ_{i}, σ^{2}_{i}), i = 1, ⋯, k, independent random variables that have probability distributions related to the chisquared distribution:
Name  Statistic 

chisquared distribution  
noncentral chisquared distribution  
chi distribution  
noncentral chi distribution 
The chisquared distribution is also often encountered in Magnetic Resonance Imaging .^{[15]}
Table of χ^{2} value vs pvalue
The pvalue is the probability of observing a test statistic at least as extreme in a chisquared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the pvalue. The table below gives a number of pvalues matching to χ^{2} for the first 10 degrees of freedom.
A low pvalue indicates greater statistical significance, i.e. greater confidence that the observed deviation from the null hypothesis is significant. A pvalue of 0.05 is often used as a brightline cutoff between significant and notsignificant results.
Degrees of freedom (df)  χ^{2} value^{[16]}  

1

0.004  0.02  0.06  0.15  0.46  1.07  1.64  2.71  3.84  6.64  10.83 
2

0.10  0.21  0.45  0.71  1.39  2.41  3.22  4.60  5.99  9.21  13.82 
3

0.35  0.58  1.01  1.42  2.37  3.66  4.64  6.25  7.82  11.34  16.27 
4

0.71  1.06  1.65  2.20  3.36  4.88  5.99  7.78  9.49  13.28  18.47 
5

1.14  1.61  2.34  3.00  4.35  6.06  7.29  9.24  11.07  15.09  20.52 
6

1.63  2.20  3.07  3.83  5.35  7.23  8.56  10.64  12.59  16.81  22.46 
7

2.17  2.83  3.82  4.67  6.35  8.38  9.80  12.02  14.07  18.48  24.32 
8

2.73  3.49  4.59  5.53  7.34  9.52  11.03  13.36  15.51  20.09  26.12 
9

3.32  4.17  5.38  6.39  8.34  10.66  12.24  14.68  16.92  21.67  27.88 
10

3.94  4.87  6.18  7.27  9.34  11.78  13.44  15.99  18.31  23.21  29.59 
P value (Probability)

0.95  0.90  0.80  0.70  0.50  0.30  0.20  0.10  0.05  0.01  0.001 
See also
{{#invoke:Portalportal}} Template:Colbegin
 Cochran's theorem
 Fdistribution
 Fisher's method for combining independent tests of significance
 Gamma distribution
 Generalized chisquared distribution
 Noncentral chisquared distribution
 Hotelling's Tsquared distribution
 Pearson's chisquared test
 Student's tdistribution
 Wilks' lambda distribution
 Wishart distribution
References
 ↑ Template:Abramowitz Stegun ref
 ↑ NIST (2006). Engineering Statistics Handbook  ChiSquared Distribution
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ F. R. Helmert, "Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen", Zeitschrift für Mathematik und Physik 21, 1876, S. 102–219
 ↑ R. L. Plackett, Karl Pearson and the ChiSquared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics.
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ Chisquared distribution, from MathWorld, retrieved Feb. 11, 2009
 ↑ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 9780387346571
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica, [1]
 ↑ ChiSquared Test Table B.2. Dr. Jacqueline S. McLaughlin at The Pennsylvania State University. In turn citing: R.A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV
Further reading
External links
 {{#invoke:citation/CS1citation
CitationClass=citation }}
 Calculator for the pdf, cdf and quantiles of the chisquared distribution
 Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
 Course notes on ChiSquared Goodness of Fit Testing from Yale University Stats 101 class.
 Mathematica demonstration showing the chisquared sampling distribution of various statistics, e.g. Σx², for a normal population
 Simple algorithm for approximating cdf and inverse cdf for the chisquared distribution with a pocket calculator
 REDIRECT Template:Probability distributions