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
|curtosis:||c = 12/N + 3|
In the limit the distribution approaches the normal distribution with mean N and variance 2N. 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.