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# Chi-Squared Distribution Random Number Generator

The chi-squared distribution with *k* degrees of freedom, denoted χ^{2}_{k} is the distribution of the sum of the squares of *k* normally distributed random variables (with mean 0 and variance 1). Chi-squared distributions are often encountered when studying multiple data sets whose elements come from a normal distribution. Statistical hypothesis testing is another application. You can use the calculator below to generate random variables from a χ^{2}_{k} distribution.

The PDF of the chi-squared distribution is

The theoretical mean, median, mode, variance, and skewness of the chi-squared distribution depend solely on the value of

*k*. The equations are

mean =

*k*

mode = 0 if

*k*< 3,

*k*-2 otherwise

variance = 2

*k*

skew = sqrt(8/

*k*)

There is no closed form of the median of the χ

^{2}

_{k}distribution, Med(

*k*). However, it can be approximated with the following equation:

Med(

*k*) = 0.455 for

*k*= 1,

Med(

*k*) = 1.386 for

*k*= 2,

Med(

*k*) =

*k*(1 - 0.224/

*k*)³ for

*k*> 2

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