{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,| cdf =$\backslash frac\{\backslash gamma(k/2,x/2)\}\{\backslash Gamma(k/2)\}\backslash ,$| mean =$k\backslash ,$| median =approximately $k-2/3\backslash ,$| mode =$k-2\backslash ,$ if $k\backslash geq\; 2\backslash ,$| variance =$2\backslash ,k\backslash ,$| skewness =$\backslash sqrt\{8/k\}\backslash ,$| kurtosis =$12/k\backslash ,$| entropy =$\backslash frac\{k\}\{2\}\backslash !+\backslash !\backslash ln(2\backslash Gamma(k/2))\backslash !+\backslash !(1\backslash !-\backslash !k/2)\backslash psi(k/2)$| mgf =$(1-2\backslash ,t)^\{-k/2\}$ for | char =$(1-2\backslash ,i\backslash ,t)^\{-k/2\}\backslash ,$ }} In probability theory and statistics, the chi-square distribution (also chi-squared or χ²  distribution) is one of the theoretical probability distributions most widely used 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 $X\_i$ are k independent, normally distributed random variables with means $\backslash mu\_i$ and variances $\backslash sigma\_i^2$, then the random variable

$Z\; =\; \backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2$

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

$Z\backslash sim\backslash chi^2\_k.\backslash ,$

The chi-square distribution has one parameter: $k$ - a positive integer which specifies the number of degrees of freedom (i.e. the number of $X\_i$)

The chi-square distribution is a special case of the gamma distribution.

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

Properties

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The chi-square probability density function is

$$

f(x;k)= \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}

where $x\; \backslash ge\; 0$ and $f(x;k)\; =\; 0$ for $x\; \backslash le\; 0$. Here $\backslash Gamma$ denotes the Gamma function. The cumulative distribution function is:

$F(x;k)=\backslash frac\{\backslash gamma(k/2,x/2)\}\{\backslash Gamma(k/2)\}\backslash ,$

where $\backslash gamma(k,z)$ is the incomplete Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available (see the External links below for online versions), and the function is included in many spreadsheets (for example OpenOffice.org calc or Microsoft Excel) and all statistical packages.

If $p$ independent linear homogeneous constraints are imposed on these variables, the distribution of $X$ conditional on these constraints is $\backslash chi^2\_\{k-p\}$, justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is

$\backslash phi(t;k)=(1-2it)^\{-k/2\}.\backslash ,$

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square 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 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 variables divided by their respective degrees of freedom.

The normal approximation

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

Fisher showed that $\backslash sqrt\{2X\}$ is approximately normally distributed with mean $\backslash sqrt\{2k-1\}$ and unit variance.

Wilson and Hilferty showed in 1931 that $\backslash sqrt*\{X/k\}$ is approximately normally distributed with mean $1-2/(9k)$ and variance $2/(9k)$.

The expected value of a random variable having chi-square distribution with $k$ degrees of freedom is $k$ and the variance is $2k$. The median is given approximately by

$k-\backslash frac\{2\}\{3\}+\backslash frac\{4\}\{27k\}-\backslash frac\{8\}\{729k^2\}.$

Note that 2 degrees of freedom leads to an exponential distribution.

The information entropy is given by

$$

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).

where $\backslash psi(x)$ is the Digamma function.

Related distributions

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• $X\; \backslash sim\; \backslash mathrm\{Exponential\}(\backslash lambda\; =\; 2)$ is an exponential distribution (where λ is a survival parameter) if $X\; \backslash sim\; \backslash chi\_2^2$ (with 2 degrees of freedom).
• $Y\; \backslash sim\; \backslash chi\_k^2$ is a chi-square distribution if $Y\; =\; \backslash sum\_\{m=1\}^k\; X\_m^2$ for $X\_i\; \backslash sim\; N(0,1)$ independent that are normally distributed. If the $X\_i\backslash sim\; N(\backslash mu\_i,1)$ have nonzero means, then $Y\; =\; \backslash sum\_\{m=1\}^k\; X\_m^2$ is drawn from a noncentral chi-square distribution.
• $Y\; \backslash sim\; \backslash mathrm\{F\}(\backslash nu\_1,\; \backslash nu\_2)$ is an F-distribution if $Y\; =\; (X\_1\; /\; \backslash nu\_1)/(X\_2\; /\; \backslash nu\_2)$ where $X\_1\; \backslash sim\; \backslash chi\_\{\backslash nu\_1\}^2$ and $X\_2\; \backslash sim\; \backslash chi\_\{\backslash nu\_2\}^2$ are independent with their respective degrees of freedom.
• $Y\; \backslash sim\; \backslash chi^2(\backslash bar\{\backslash nu\})$ is a chi-square distribution if $Y\; =\; \backslash sum\_\{m=1\}^N\; X\_m$ where $X\_m\; \backslash sim\; \backslash chi^2(\backslash nu\_m)$ are independent and $\backslash bar\{\backslash nu\}\; =\; \backslash sum\_\{m=1\}^N\; \backslash nu\_m$.
• if $X$ is chi-square distributed, then $\backslash sqrt\{X\}$ is chi distributed.
• if $X\_1,\; \backslash dots,\; X\_n$ are i.i.d. $N(\backslash mu,\backslash sigma^2)$ random variables, then $\backslash sum\_\{i=1\}^n(X\_i\; -\; \backslash bar\; X)^2\; \backslash sim\; \backslash sigma^2\; \backslash chi^2\_\{n-1\}$ where $\backslash bar\; X\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; X\_i$.

Various chi and chi-square distributions

Name	Statistic
chi-square distribution	$\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2$
noncentral chi-square distribution	$\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i\}\{\backslash sigma\_i\}\backslash right)^2$
chi distribution	$\backslash sqrt\{\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$
noncentral chi distribution	$\backslash sqrt\{\backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{X\_i\}\{\backslash sigma\_i\}\backslash right)^2\}$

See also

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• Cochran's theorem
• Inverse-chi-square distribution

External links

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• SixSigmaFirst, On line tutorials for Six Sigma and Statistics
• On-line calculator for the significance of chi-square, in Richard Lowry's statistical website at Vassar College.
• Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
• Chi-Square Calculator for critical values of Chi-Square in R. Webster West's applet website at University of South Carolina

Continuous distributions

Chi-Quadrat-Verteilung | Distribución chi-cuadrado | Loi du χ² | Variabile casuale chi quadro | Chi-kwadraatverdeling | カイ二乗分布 | Rozkład chi kwadrat | Sebaran chi-kuadrat | Khii toiseen -jakauma | Chitvåfördelning | Ki-kare dağılımı | 卡方分佈