View source for Chi-square distribution

{{Otheruses4|the mathematics of the chi-square distribution|its uses in statistics|chi-square test}}

{{Probability distribution|
  name       =chi-square|
  type       =density|
  pdf_image  =[[Image:chi-square_distributionPDF.png|325px]]|
  cdf_image  =[[Image:chi-square_distributionCDF.png|325px]]|
  parameters =<math>k > 0\,</math> degrees of freedom|
  support    =<math>x \in [0; +\infty)\,</math>|
  pdf        =<math>\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>|
  cdf        =<math>\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,</math>|
  mean       =<math>k\,</math>|
  median     =approximately <math>k-2/3\,</math>|
  mode       =<math>k-2\,</math> if <math>k\geq 2\,</math>|
  variance   =<math>2\,k\,</math>|
  skewness   =<math>\sqrt{8/k}\,</math>|
  kurtosis   =<math>12/k\,</math>|
  entropy    =<math>\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)</math>|
  mgf        =<math>(1-2\,t)^{-k/2}</math> for <math>2\,t<1\,</math>|
  char       =<math>(1-2\,i\,t)^{-k/2}\,</math>
}}


In [[probability theory]] and [[statistics]], the '''chi-square distribution''' (also '''chi-squared''' or '''<math>\chi^2</math>&nbsp; distribution''') is one of the most widely used [[theoretical]] [[probability distribution]]s in [[inferential statistics]], i.e. in [[statistical significance]] tests.  It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the [[null hypothesis]] is true.

If <math>X_i</math> are ''k'' [[statistical independence|independent]], [[normal distribution|normally distributed]] random variables with [[mean]] 0 and [[variance]] 1, then the random variable

:<math>Q = \sum_{i=1}^k X_i^2</math>

is distributed according to the  chi-square distribution. This is usually written

:<math>Q\sim\chi^2_k.\,</math>

The chi-square distribution has one parameter: <math>k</math> - a positive integer that specifies the number of [[degrees of freedom (statistics)|degrees of freedom]] (i.e. the number of <math>X_i</math>)

The chi-square distribution is a special case of the [[gamma distribution#Related distributions|gamma distribution]].

The best-known situations in which the chi-square distribution is used are the common [[chi-square test]]s for [[goodness of fit]] of an observed distribution to a theoretical one, and of the [[statistical independence|independence]] of two criteria of classification of [[qualitiative data analysis|qualitative data]].  However, many other statistical tests lead to a use of this distribution. One example is [[Friedman test|Friedman's analysis of variance by ranks]].  

==Characteristics==
===Probability density function===
A [[probability density function]] of the chi-square distribution is 

:<math>
f(x;k)=
\begin{cases}\displaystyle
\frac{1}{2^{k/2}\Gamma(k/2)}\,x^{k/2 - 1} e^{-x/2}&\text{for }x>0,\\
0&\text{for }x\le0,
\end{cases}
</math>

where <math>\Gamma</math> denotes the [[Gamma function]], which takes [[particular values of the Gamma function|particular values at the half-integers]].

===Cumulative distribution function===
Its [[cumulative distribution function]] is:

:<math>F(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)} = P(k/2, x/2)</math>

where <math>\gamma(k,z)</math> is the [[incomplete Gamma function|lower incomplete Gamma function]] and <math>P(k, z)</math> is the [[regularized Gamma function]].

Tables of this distribution &mdash; usually in its cumulative form &mdash; are widely available and the function is included in many [[spreadsheet]]s and all [[List of statistical packages|statistical packages]].

===Characteristic function===
The [[Characteristic function (probability theory)|characteristic function]] of the Chi-square distribution is

:<math>\phi(t;k)=(1-2it)^{-k/2}.\,</math>

==Properties==

The chi-square distribution has numerous applications in inferential [[statistics]], for instance in [[chi-square test]]s and in estimating [[variance]]s.
It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a [[linear regression|regression]] line via its role in [[Student's t-distribution]].
It enters all [[analysis of variance]] problems via its role in the [[F-distribution]], which is the distribution of the ratio of two independent chi-squared [[random variable]]s divided by their respective degrees of freedom.

====Normal approximation====

If <math>X\sim\chi^2_k</math>, then as <math>k</math> tends to infinity, the distribution of <math>X</math> tends to normality.
However, the tendency is  slow (the skewness is <math>\sqrt{8/k}</math> and the kurtosis excess is <math>12/k</math>) and two transformations are commonly considered, each of which approaches normality faster than <math>X</math> itself:

Fisher emprically showed that <math>\sqrt{2X}</math> is approximately normally distributed with mean <math>\sqrt{2k-1}</math> and unit variance. It is possible to arrive at the same normal approximation result by using moment matching. To see this, consider the mean and the variance of a Chi-distributed random variable <math>z=\sqrt{X}</math>, which are given by <math>\mu_z= \sqrt{2}
\frac{\Gamma\left(k/2+1/2\right)}{\Gamma\left(k/2 \right)}</math> and
<math>\sigma_z^2= k-\mu_z^2</math>, where <math>\Gamma(\cdot)</math> is the Gamma
function. The particular ratio of the Gamma
functions in <math>\mu_z </math> has the following series expansion [http://www.umiacs.umd.edu/users/volkan/SSP07.pdf]: <math>\frac{\Gamma\left(N+1/2\right)}{\Gamma\left(N
\right)}=\sqrt{N}\left(1-\frac{1}{8N}+
\frac{1}{128N^2}+\frac{5}{1024N^3}-\frac{21}{32768N^4}+\ldots\right).</math>
When <math>N\gg 1</math>,  this ratio can be approximated as
follows:
<math> \frac{\Gamma\left(N+1/2\right)}{\Gamma\left(N
\right)}\approx\sqrt{N}\left(1-\frac{1}{8N}\right)\approx\sqrt{N}\left(1-\frac{1}{4N}\right)^{0.5}=\sqrt{N-1/4}.
</math>
Then, simple moment matching results in the following approximation of <math>z</math>:
<math> z\sim{\mathcal N}\left(\sqrt{k-1/2},
\frac{1}{2}\right)</math>, from which it follows that <math> \sqrt{2X}\sim{\mathcal N}\left(\sqrt{2k-1},
1\right)</math>.

Wilson and Hilferty showed in [[1931]] that <math>\sqrt[3]{X/k}</math> is approximately normally distributed with mean <math>1-2/(9k)</math> and variance <math>2/(9k)</math>.

The [[expected value]] of a random variable having chi-square distribution with <math>k</math> degrees of freedom is <math>k</math> and the [[variance]] is <math>2k</math>.
The median is given approximately by

:<math>k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.</math>

Note that 2 degrees of freedom lead to an [[exponential distribution]].

===Information entropy===
The [[information entropy]] is given by

:<math>
H
=
\int_{-\infty}^\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(k/2).
</math>

where <math>\psi(x)</math> is the [[Digamma function]].

==Related distributions==

*<math>X \sim \mathrm{Exponential}(\lambda = \frac{1}{2})</math> <!-- Used to be 2; might be a typo. If one looks at the mgf the mistake is obvious. ~~~~icefox -->is an [[exponential distribution]] if <math>X \sim \chi_2^2</math> (with 2 [[degrees of freedom (statistics)|degrees of freedom]]).
*<math>Y \sim \chi_k^2</math> is a chi-square distribution if <math>Y = \sum_{m=1}^k X_m^2</math> for <math>X_i \sim N(0,1)</math> [[statistical independence|independent]] that are [[normal distribution|normally distributed]].  
*If the <math>X_i\sim N(\mu_i,1)</math> have nonzero means, then <math>Y = \sum_{m=1}^k X_m^2</math> is drawn from a [[noncentral chi-square distribution]].
*The chi-square distribution <math>X \sim \chi^2_\nu</math> is a special case of the [[gamma distribution]], in that <math>X \sim \textrm{Gamma}(\tfrac{\nu}{2}, 2)</math>.
*<math>Y \sim \mathrm{F}(\nu_1, \nu_2)</math> is an [[F-distribution]] if <math>Y = \frac{X_1 / \nu_1}{X_2 / \nu_2}</math> where <math>X_1 \sim \chi_{\nu_1}^2</math> and <math>X_2 \sim \chi_{\nu_2}^2</math> are independent with their respective degrees of freedom.
*<math>Y \sim \chi^2(\bar{\nu})</math> is a chi-square distribution if <math>Y = \sum_{m=1}^N X_m</math> where <math>X_m \sim \chi^2(\nu_m)</math> are independent and <math>\bar{\nu} = \sum_{m=1}^N \nu_m</math>.
*if <math>X</math> is chi-square distributed, then <math>\sqrt{X}</math> is [[chi distribution|chi distributed]].
*in particular, if <math>X \sim \chi_2^2</math> (chi-square with 2 degrees of freedom), then <math>\sqrt{X}</math> is [[Rayleigh distribution|Rayleigh distributed]].
*if <math>X_1, \dots, X_n</math> are [[independent identically-distributed random variables|i.i.d.]] <math>N(\mu,\sigma^2)</math> [[random variable]]s, then <math>\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}</math> where <math>\bar X = \frac{1}{n} \sum_{i=1}^n X_i</math>.
*if <math>X \sim \mathrm{SkewLogistic}(\frac{1}{2})\,</math>, then <math>\mathrm{log}(1 + e^{-X}) \sim  \chi_2^2\,</math>

<center>
{| class="wikitable" align="center"
|+ '''Various chi and chi-square distributions'''
|-
! Name !! Statistic
|-
| chi-square distribution || <math>\sum_{i=1}^k \frac{\left(X_i-\mu_i\right)^2}{\sigma_i}</math>
|-
| [[noncentral chi-square distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| [[chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
</center>

==See also==
*[[Cochran's theorem]]
*[[Inverse-chi-square distribution]]
*[[Degrees of freedom (statistics)]]
*[[Fisher's method]] for combining [[Statistical independence|independent]] [[test]]s of [[significance]]
*[[Noncentral chi-square distribution]]

==External links== 
*[http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm Course notes on Chi-Square Goodness of Fit Testing] from [[Yale University]] Stats 101 class. Example includes hypothesis testing and parameter estimation.
*[http://faculty.vassar.edu/lowry/tabs.html#csq On-line calculator for the significance of chi-square], in Richard Lowry's statistical website at [[Vassar College]].
*[http://www.vias.org/simulations/simusoft_distcalc.html Distribution Calculator] Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
*[http://www.stat.sc.edu/~west/applets/chisqdemo.html Chi-Square Calculator for critical values of Chi-Square] in R. Webster West's applet website at University of South Carolina
*[http://graphpad.com/quickcalcs/chisquared2.cfm Chi-Square Calculator from GraphPad]
{{ProbDistributions|Chi-square distribution}}

[[Category:Continuous distributions]]

[[ca:Distribució khi quadrat]]
[[cs:Χ² rozdělení]]
[[de:Chi-Quadrat-Verteilung]]
[[es:Distribución chi-cuadrado]]
[[fr:Loi du χ²]]
[[it:Variabile casuale chi quadro]]
[[nl:Chi-kwadraatverdeling]]
[[ja:カイ二乗分布]]
[[pl:Rozkład chi kwadrat]]
[[pt:Chi-quadrado]]
[[ru:Распределение хи-квадрат]]
[[simple:Chi-square distribution]]
[[su:Sebaran chi-kuadrat]]
[[fi:Khii toiseen -jakauma]]
[[sv:Chitvåfördelning]]
[[tr:Ki-kare dağılımı]]
[[zh:卡方分佈]]

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