The random variable (chi-square) follows the probability density of the distribution with *N* degrees of freedom

where is Euler's Gamma function. The

distribution has the properties

mean: | , |

variance: | , |

skewness: | |

curtosis: | c = 12/N + 3 |

In the limit the distribution approaches the normal distribution with mean *N* and variance 2*N*. For an *N*-independent test (e.g. comparing 's with different *N*) one can use the quantity

however, the expression

is usually preferred, as it approaches standard normal behaviour faster as *N* increases.

Rudolf K. Bock, 7 April 1998