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Chi-squared distribution
Probability distribution and special case of gamma distribution / From Wikipedia, the free encyclopedia
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In probability theory and statistics, the chi-squared distribution (also chi-square or -distribution) with
degrees of freedom is the distribution of a sum of the squares of
independent standard normal random variables.
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The chi-squared distribution is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if
then
(where
is the shape parameter and
the scale parameter of the gamma distribution) and
.
The scaled chi-squared distribution is a reparametrization of the gamma distribution and the univariate Wishart distribution. Specifically if
then
and
.
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.[2][3][4][5] 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.