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Chi-square distribution

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The title of this article should be chi-square distribution or χ2 distribution. The initial letter is capitalized due to technical restrictions.

Probability density function
Chi-square distributionPDF
Cumulative distribution function
Chi-square distributionCDF
Parameters k > 0\, degrees of freedom
Support x \in [0; +\infty)\,
pdf \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,
cdf \frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,
Mean k\,
Median approximately k-2/3\,
Mode k-2\, if k\geq 2\,
Variance 2\,k\,
Skewness \sqrt{8/k}\,
Kurtosis 12/k\,
Entropy \frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)
mgf (1-2\,t)^{-k/2} for 2\,t<1\,
Char. func. (1-2\,i\,t)^{-k/2}\,

In probability theory and statistics, the chi-square distribution (also chi-squared distribution), 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 proved 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 \mu_i and variances \sigma_i^2, then the statistic

Z = \sum_{i=1}^k \left(\frac{x_i-\mu_i}{\sigma_i}\right)^2

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


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, for example Friedman's analysis of variance by ranks.


The chi-square probability density function is

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

where x \ge 0 and f_k(x) = 0 for x \le 0. Here \Gamma denotes the Gamma function. The cumulative distribution function is:


where \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 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 \chi^2_{k-p}, justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is


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 approximationEdit

If X\sim\chi^2_k, then as k tends to infinity, the distribution of X tends to normality. However, the tendency is slow (the skewness is \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 \sqrt{2X} is approximately normally distributed with mean \sqrt{2k-1} and unit variance.

Wilson and Hilferty showed in 1931 that \sqrt[3]{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


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

The information entropy is given by:

\int_{-\infty}^\infty f(x;k)\ln(f(x;k)) dx
  2 \Gamma
\left(1 - \frac{k}{2}\right)

where \psi(x) is the Digamma function.

Related distributionsEdit

Various chi and chi-square distributions
Name Statistic
chi-square distribution \sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2
noncentral chi-square distribution \sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2
chi distribution \sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}
noncentral chi distribution \sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}

See alsoEdit

External linksEdit

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