In probability theory and statistics, the chisquared distribution (also chisquare or χ^{2}distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chisquare distribution is a special case of the gamma distribution and 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 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, such as Friedman's analysis of variance by ranks.
chisquared  

Probability density function


Cumulative distribution function


Notation  or 
Parameters  (known as "degrees of freedom") 
Support  if , otherwise 
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  ^{[1]} 
PGF 
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).
The chisquared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chisquared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others.
It is also a component of the definition of the tdistribution and the Fdistribution used in ttests, analysis of variance, and regression analysis.
The primary reason that the chisquared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the tstatistic in a ttest. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chisquared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chisquared distribution could be used.
Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution. Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z^{2}. The distribution of the random variable Q is an example of a chisquared distribution: The subscript 1 indicates that this particular chisquared distribution is constructed from only 1 standard normal distribution. A chisquared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chisquared distribution. Just as extreme values of the normal distribution have low probability (and give small pvalues), extreme values of the chisquared distribution have low probability.
An additional reason that the chisquared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT).^{[6]} LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chisquared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chisquared approximation for small sample size. Similarly, in analyses of contingency tables, the chisquared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey show that the exact binomial test is always more powerful than the normal approximation.^{[7]}
Lancaster^{[8]} shows the connections among the binomial, normal, and chisquared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable
where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.
Squaring both sides of the equation gives
Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to
The expression on the right is of the form that Pearson would generalize to the form:
where
In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chisquared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chisquared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chisquared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution ^{[8]}
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 chisquare distribution is
where 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 is the lower incomplete gamma function and is the regularized gamma function.
In a special case of k = 2 this function has a simple form:
and the integer recurrence of the gamma function makes it easy to compute 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 i.i.d. chisquared variables of degree is distributed according to a gamma distribution with shape and scale 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 ).
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 momentgenerating function of the sufficient statistic.
The moments about zero of a chisquared distribution with k degrees of freedom are given by^{[10]}^{[11]}^{[12]}
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.^{[13]} Specifically, if X ~ χ^{2}(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},^{[14]} as the logarithm removes much of the asymmetry.^{[15]} Other functions of the chisquared distribution converge more rapidly to a normal distribution. Some examples are:
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.
If is a positivesemidefinite covariance matrix with strictly positive diagonal entries, then for and a random vector independent of such that and it holds that
^{[15]}
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 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.^{[17]}
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 is a special case of the gamma distribution, in that using the rate parameterization of the gamma distribution (or 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 , then is an exponential distribution.
The Erlang distribution is also a special case of the gamma distribution and thus we also have that if 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.^{[18]}
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 cutoff between significant and notsignificant results.
Degrees of freedom (df)  χ^{2} value^{[19]}  

1  0.004  0.02  0.06  0.15  0.46  1.07  1.64  2.71  3.84  6.63  10.83 
2  0.10  0.21  0.45  0.71  1.39  2.41  3.22  4.61  5.99  9.21  13.82 
3  0.35  0.58  1.01  1.42  2.37  3.66  4.64  6.25  7.81  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 
These values can be calculated evaluating the quantile function (also known as “inverse CDF” or “ICDF”) of the chisquared distribution;^{[20]} e. g., the χ^{2} ICDF for p = 0.05 and df = 7 yields 14.06714 ≈ 14.07 as in the table above. In Python, this can be calculated using SciPy via scipy.stats.chi2.isf(0.05, 7)
^{[21]}
This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875–6,^{[22]}^{[23]} where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmert'sche ("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 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 −½χ^{2} for what would appear in modern notation as −½x^{T}Σ^{−1}x (Σ being the covariance matrix).^{[24]} 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.^{[22]}
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