Chi-squared distribution

The chi-squared distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables
• Wald test
• Score test

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason for which the chi-squared distribution is extensively used in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. 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 chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Suppose that ${\displaystyle Z}$ is a random variable sampled from the standard normal distribution, where the mean is ${\displaystyle 0}$ and the variance is ${\displaystyle 1}$: ${\displaystyle Z\sim N(0,1)}$. Now, consider the random variable ${\displaystyle Q=Z^{2}}$. The distribution of the random variable ${\displaystyle Q}$ is an example of a chi-squared distribution: ${\displaystyle \ Q\ \sim \ \chi _{1}^{2}}$. The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared 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. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT).[6] LRTs have several desirable properties; in particular, simple LRTs commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for a small sample size, and it is preferable to use Fisher's exact test. Ramsey shows that the exact binomial test is always more powerful than the normal approximation.[7]

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows.[8] 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

${\displaystyle \chi ={m-Np \over {\sqrt {Npq}}}}$

where ${\displaystyle m}$ is the observed number of successes in ${\displaystyle N}$ trials, where the probability of success is ${\displaystyle p}$, and ${\displaystyle q=1-p}$.

Squaring both sides of the equation gives

${\displaystyle \chi ^{2}={(m-Np)^{2} \over Npq}}$

Using ${\displaystyle N=Np+N(1-p)}$, ${\displaystyle N=m+(N-m)}$, and ${\displaystyle q=1-p}$, this equation can be rewritten as

${\displaystyle \chi ^{2}={(m-Np)^{2} \over Np}+{(N-m-Nq)^{2} \over Nq}}$

The expression on the right is of the form that Karl Pearson would generalize to the form

${\displaystyle \chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}}$

where

${\displaystyle \chi ^{2}}$ = Pearson's cumulative test statistic, which asymptotically approaches a ${\displaystyle \chi ^{2}}$ distribution; ${\displaystyle O_{i}}$ = the number of observations of type ${\displaystyle i}$; ${\displaystyle E_{i}=Np_{i}}$ = the expected (theoretical) frequency of type ${\displaystyle i}$, asserted by the null hypothesis that the fraction of type ${\displaystyle i}$ in the population is ${\displaystyle p_{i}}$; and ${\displaystyle n}$ = the number of cells in the table.[9]

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large ${\displaystyle n}$). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value. 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 degenerate multivariate normal approximation to the multinomial distribution (the numbers in each category add up to the total sample size, which is considered fixed). Pearson showed that the chi-squared distribution arose from such a multivariate normal approximation to the multinomial distribution, taking careful account of the statistical dependence (negative correlations) between numbers of observations in different categories.[8]

The probability density function (pdf) of the chi-squared distribution is

${\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\\0,&{\text{otherwise}}.\end{cases}}}$

where ${\textstyle \Gamma (k/2)}$ denotes the gamma function, which has closed-form values for integer ${\displaystyle k}$.

For derivations of the pdf in the cases of one, two and ${\displaystyle k}$ degrees of freedom, see Proofs related to chi-squared distribution.

Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (${\displaystyle k=10}$)

Its cumulative distribution function is:

${\displaystyle F(x;\,k)={\frac {\gamma ({\frac {k}{2}},\,{\frac {x}{2}})}{\Gamma ({\frac {k}{2}})}}=P\left({\frac {k}{2}},\,{\frac {x}{2}}\right),}$

where ${\displaystyle \gamma (s,t)}$ is the lower incomplete gamma function and ${\textstyle P(s,t)}$ is the regularized gamma function.

In a special case of ${\displaystyle k=2}$ this function has the simple form:

${\displaystyle F(x;\,2)=1-e^{-x/2}}$

which can be easily derived by integrating ${\displaystyle f(x;\,2)={\frac {1}{2}}e^{-x/2}}$ directly. The integer recurrence of the gamma function makes it easy to compute ${\displaystyle F(x;\,k)}$ for other small, even ${\displaystyle k}$.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.

Letting ${\displaystyle z\equiv x/k}$, Chernoff bounds on the lower and upper tails of the CDF may be obtained.[10] For the cases when ${\displaystyle 0<z<1}$ (which include all of the cases when this CDF is less than half): ${\displaystyle F(zk;\,k)\leq (ze^{1-z})^{k/2}.}$

The tail bound for the cases when ${\displaystyle z>1}$, similarly, is

${\displaystyle 1-F(zk;\,k)\leq (ze^{1-z})^{k/2}.}$

For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-squared distribution.

If ${\displaystyle Z_{1},...,Z_{n}}$ are independent identically distributed (i.i.d.), standard normal random variables, then ${\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}\sim \chi _{n-1}^{2}}$ where ${\displaystyle {\bar {Z}}={\frac {1}{n}}\sum _{t=1}^{n}Z_{t}.}$

A direct and elementary proof is as follows: Let ${\displaystyle Z\sim {\mathcal {N}}({\bar {0}},1\!\!1)}$ be a vector of ${\displaystyle n}$ independent normally distributed random variables, and ${\displaystyle {\bar {Z}}}$ their average. Then ${\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}~=~\sum _{t=1}^{n}Z_{t}^{2}-n{\bar {Z}}^{2}~=~Z^{\top }[1\!\!1-{\textstyle {\frac {1}{n}}}{\bar {1}}{\bar {1}}^{\top }]Z~=:~Z^{\top }\!MZ}$ where ${\displaystyle 1\!\!1}$ is the identity matrix and ${\displaystyle {\bar {1}}}$ the all ones vector. ${\displaystyle M}$ has one eigenvector ${\displaystyle b_{1}:={\bar {1}}}$ with eigenvalue ${\displaystyle 0}$, and ${\displaystyle n-1}$ eigenvectors ${\displaystyle b_{2},...,b_{n}}$ (all orthogonal to ${\displaystyle b_{1}}$) with eigenvalue ${\displaystyle 1}$, which can be chosen so that ${\displaystyle Q:=(b_{1},...,b_{n})}$ is an orthogonal matrix. Since also ${\displaystyle X:=Q^{\top }\!Z\sim {\mathcal {N}}({\bar {0}},Q^{\top }\!1\!\!1Q)={\mathcal {N}}({\bar {0}},1\!\!1)}$, we have ${\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}~=~Z^{\top }\!MZ~=~X^{\top }\!Q^{\top }\!MQX~=~X_{2}^{2}+...+X_{n}^{2}~\sim ~\chi _{n-1}^{2},}$ which proves the claim.

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if ${\displaystyle X_{i},i={\overline {1,n}}}$ are independent chi-squared variables with ${\displaystyle k_{i}}$, ${\displaystyle i={\overline {1,n}}}$ degrees of freedom, respectively, then ${\displaystyle Y=X_{1}+\cdots +X_{n}}$ is chi-squared distributed with ${\displaystyle k_{1}+\cdots +k_{n}}$ degrees of freedom.

The sample mean of ${\displaystyle n}$ i.i.d. chi-squared variables of degree ${\displaystyle k}$ is distributed according to a gamma distribution with shape ${\displaystyle \alpha }$ and scale ${\displaystyle \theta }$ parameters:

${\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim \operatorname {Gamma} \left(\alpha =n\,k/2,\theta =2/n\right)\qquad {\text{where }}X_{i}\sim \chi ^{2}(k)}$

Asymptotically, given that for a scale parameter ${\displaystyle \alpha }$ going to infinity, a Gamma distribution converges towards a normal distribution with expectation ${\displaystyle \mu =\alpha \cdot \theta }$ and variance ${\displaystyle \sigma ^{2}=\alpha \,\theta ^{2}}$, the sample mean converges towards:

${\displaystyle {\overline {X}}\xrightarrow {n\to \infty } N(\mu =k,\sigma ^{2}=2\,k/n)}$

Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-squared variable of degree ${\displaystyle k}$ the expectation is ${\displaystyle k}$ , and its variance ${\displaystyle 2\,k}$ (and hence the variance of the sample mean ${\displaystyle {\overline {X}}}$ being ${\displaystyle \sigma ^{2}={\frac {2k}{n}}}$).

The differential entropy is given by

${\displaystyle h=\int _{0}^{\infty }f(x;\,k)\ln f(x;\,k)\,dx={\frac {k}{2}}+\ln \left[2\,\Gamma \left({\frac {k}{2}}\right)\right]+\left(1-{\frac {k}{2}}\right)\,\psi \!\left({\frac {k}{2}}\right),}$

where ${\displaystyle \psi (x)}$ is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate ${\displaystyle X}$ for which ${\displaystyle \operatorname {E} (X)=k}$ and ${\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)}$ are fixed. Since the chi-squared 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 chi-squared distribution with ${\displaystyle k}$ degrees of freedom are given by[11][12]

${\displaystyle \operatorname {E} (X^{m})=k(k+2)(k+4)\cdots (k+2m-2)=2^{m}{\frac {\Gamma \left(m+{\frac {k}{2}}\right)}{\Gamma \left({\frac {k}{2}}\right)}}.}$

The cumulants are readily obtained by a power series expansion of the logarithm of the characteristic function:

${\displaystyle \kappa _{n}=2^{n-1}(n-1)!\,k}$

The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart[13] bounds are:

${\displaystyle \operatorname {P} (X-k\geq 2{\sqrt {kx}}+2x)\leq \exp(-x)}$

${\displaystyle \operatorname {P} (k-X\geq 2{\sqrt {kx}})\leq \exp(-x)}$

One consequence is that, if ${\displaystyle v\sim N(0,1)^{n}}$ is a gaussian random vector in ${\displaystyle \mathbb {R} ^{n}}$, then as the dimension ${\displaystyle n}$ grows, the squared length of the vector is concentrated tightly around ${\displaystyle n}$ with a width ${\displaystyle n^{1/2+\alpha }}$:

$${\displaystyle Pr(\|v\|^{2}\in [n-2n^{1/2+\alpha },n+2n^{1/2+\alpha }+2n^{\alpha }])\leq e^{-n^{\alpha }}}$$

where the exponent ${\displaystyle \alpha }$ can be chosen as any value in ${\displaystyle (0,1/2)}$.

Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.

By the central limit theorem, because the chi-squared distribution is the sum of ${\displaystyle k}$ independent random variables with finite mean and variance, it converges to a normal distribution for large ${\displaystyle k}$. For many practical purposes, for ${\displaystyle k>50}$ the distribution is sufficiently close to a normal distribution, so the difference is ignorable.[14] Specifically, if ${\displaystyle X\sim \chi ^{2}(k)}$, then as ${\displaystyle k}$ tends to infinity, the distribution of ${\displaystyle (X-k)/{\sqrt {2k}}}$ tends to a standard normal distribution. However, convergence is slow as the skewness is ${\displaystyle {\sqrt {8/k}}}$ and the excess kurtosis is ${\displaystyle 12/k}$.

The sampling distribution of ${\displaystyle \ln(\chi ^{2})}$ converges to normality much faster than the sampling distribution of ${\displaystyle \chi ^{2}}$,[15] as the logarithmic transform removes much of the asymmetry.[16]

Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

• If ${\displaystyle X\sim \chi ^{2}(k)}$ then ${\displaystyle {\sqrt {2X}}}$ is approximately normally distributed with mean ${\displaystyle {\sqrt {2k-1}}}$ and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of Johnson.[4]
• If ${\displaystyle X\sim \chi ^{2}(k)}$ then ${\displaystyle {\sqrt[{3}]{X/k}}}$ is approximately normally distributed with mean ${\displaystyle 1-{\frac {2}{9k}}}$ and variance ${\displaystyle {\frac {2}{9k}}.}$[17] This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of Johnson.[4]
  • This normalizing transformation leads directly to the commonly used median approximation ${\displaystyle k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}$ by back-transforming from the mean, which is also the median, of the normal distribution.

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance 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 ${\displaystyle X_{1},\ldots ,X_{n}}$ are chi square random variables and ${\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}}$, then a closed expression for the distribution of ${\displaystyle X=\sum _{i=1}^{n}a_{i}X_{i}}$ is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-square random variables.[19]

The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

The generalized chi-squared distribution is obtained from the quadratic form z'Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

The chi-squared distribution ${\displaystyle X\sim \chi _{k}^{2}}$ is a special case of the gamma distribution, in that ${\displaystyle X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)}$ using the rate parameterization of the gamma distribution (or ${\displaystyle X\sim \Gamma \left({\frac {k}{2}},2\right)}$ 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 ${\displaystyle X\sim \chi _{2}^{2}}$, then ${\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{2}}\right)}$ is an exponential distribution.

The Erlang distribution is also a special case of the gamma distribution and thus we also have that if ${\displaystyle X\sim \chi _{k}^{2}}$ with even ${\displaystyle k}$, then ${\displaystyle X}$ is Erlang distributed with shape parameter ${\displaystyle k/2}$ and scale parameter ${\displaystyle 1/2}$.

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared 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 p-value. A low p-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results.

The table below gives a number of p-values matching to ${\displaystyle \chi ^{2}}$ for the first 10 degrees of freedom.

Degrees of freedom (df)	${\displaystyle \chi ^{2}}$ value[21]
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 chi-squared distribution;[22] e. g., the χ² ICDF for p = 0.05 and df = 7 yields 2.1673 ≈ 2.17 as in the table above, noticing that 1 – p is the p-value from the table.