Chi-squared distribution

Chi-squared
Probability density function
Cumulative distribution function
Notation	${\displaystyle \chi ^{2}(k)\;}$ or ${\displaystyle \chi _{k}^{2}\!}$
Parameters	${\displaystyle k\in \mathbb {N} ^{*}~~}$ (known as "degrees of freedom")
Support	${\displaystyle x\in (0,+\infty )\;}$
PDF	${\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\;}$
CDF	${\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)\;}$
Mean	${\displaystyle k}$
Median	${\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}$
Mode	${\displaystyle \max(k-2,0)\;}$
Variance	${\displaystyle 2k\;}$
Skewness	${\displaystyle {\sqrt {8/k}}\,}$
Excess kurtosis	${\displaystyle {\frac {12}{k}}}$
Entropy	${\displaystyle {\begin{aligned}{\frac {k}{2}}&+\log \left(2\Gamma {\Bigl (}{\frac {k}{2}}{\Bigr )}\right)\\&\!+\left(1-{\frac {k}{2}}\right)\psi \left({\frac {k}{2}}\right)\end{aligned}}}$
MGF	${\displaystyle (1-2t)^{-k/2}{\text{ for }}t<{\frac {1}{2}}\;}$
CF	${\displaystyle (1-2it)^{-k/2}}$
PGF	${\displaystyle (1-2\ln t)^{-k/2}{\text{ for }}0<t<{\sqrt {e}}\;}$

In probability theory and statistics, the ${\displaystyle \chi ^{2}}$-distribution with ${\displaystyle k}$ degrees of freedom is the distribution of a sum of the squares of ${\displaystyle k}$ independent standard normal random variables.

The chi-squared distribution ${\displaystyle \chi _{k}^{2}}$ is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if ${\displaystyle X\sim \chi _{k}^{2}}$ then ${\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2)}$ (where ${\displaystyle \alpha }$ is the shape parameter and ${\displaystyle \theta }$ the scale parameter of the gamma distribution) and ${\displaystyle X\sim {\text{W}}_{1}(1,k)}$.

The scaled chi-squared distribution ${\displaystyle s^{2}\chi _{k}^{2}}$ is a reparametrization of the gamma distribution and the univariate Wishart distribution. Specifically if ${\displaystyle X\sim s^{2}\chi _{k}^{2}}$ then ${\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2s^{2})}$ and ${\displaystyle X\sim {\text{W}}_{1}(s^{2},k)}$.

The chi-squared distribution is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral chi-squared distribution.

The chi-squared distribution is used in the common chi-squared 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 finding the confidence interval for estimating the 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.