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Probability density function  
Cumulative distribution function  
Notation  or 

Parameters  (known as "degrees of freedom") 
Support  x ∈ [0, +∞) 
CDF  
Mean  k 
Median  
Mode  max{ k − 2, 0 } 
Variance  2k 
Skewness  
Ex. kurtosis  12 / k 
Entropy  
MGF  (1 − 2 t)^{−k/2} for t < ½ 
CF  (1 − 2 i t)^{−k/2} ^{[1]} 
Probability density function  
Cumulative distribution function  
Notation  or 

Parameters  (known as "degrees of freedom") 
Support  x ∈ [0, +∞) 
CDF  
Mean  k 
Median  
Mode  max{ k − 2, 0 } 
Variance  2k 
Skewness  
Ex. kurtosis  12 / k 
Entropy  
MGF  (1 − 2 t)^{−k/2} for t < ½ 
CF  (1 − 2 i t)^{−k/2} ^{[1]} 
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.^{[2]}^{[3]}^{[4]}^{[5]} When there is a need to contrast it with the 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.
This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875/1876,^{[6]}^{[7]} 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 (Pearson 1900), with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII). 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).^{[8]} 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.^{[6]}
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)
Further properties of the chisquared distribution can be found in the box at the upper right corner of this article.
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.
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.^{[9]} 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.
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.
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 α going to infinity, a Gamma distribution converges towards a Normal distribution with expectation μ = kθ and variance σ^{2} = kθ^{2}, the sample mean converges towards:
Note that we would have obtained the same result invoking instead the central limit theorem, noting that the expectation of the χ² is k, and its variance 2k (and hence the variance of the sample mean being 2k/n).
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.
The moments about zero of a chisquared distribution with k degrees of freedom are given by^{[10]}^{[11]}
The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
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.^{[12]} 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.
This section needs additional citations for verification. (September 2011) 
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 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,
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.
If are iid chi square random variables and , then a closed expression for the distribution of is not known. However, there exists a computationally efficient algorithm to calculate the pdf to arbitrary precision.^{[16]}
The noncentral chisquared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
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.
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.
Name  Statistic 

chisquared distribution  
noncentral chisquared distribution  
chi distribution  
noncentral chi distribution 
The chisquared distribution is also often encountered in Magnetic Resonance Imaging .^{[17]}
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^{[18]}  

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 
