The Chi-squared distribution

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The chi-squared distribution with  degrees of freedom is the distribution of a sum of the squares of  independent standard normal random variables. A consequence is that the sum of independent chi-squared variables is also chi-squared distributed. It is widely used in hypothesis testing, goodness of fit analysis or in constructing confidence intervals. It is a special case of the gamma distribution.



Distribution name

Chi-squared distribution

Common notation


 = degrees of freedom (positive integer)


Probability density function

Cumulative distribution function




(Excess) kurtosis

Characteristic function

Other comments

Its median is approximately . Its mode is . Is also known as the central chi-squared distribution (when there is a need to contrast it with the noncentral chi-squared distribution).


In the special case of  the cumulative distribution function simplifies to .


As ,  and


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “chi-squared”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “chi-squared3” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.


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