Chi-Square Distribution

  If the random variable X follows the standard normal distribution, i.e. the Gaussian distribution with zero mean and unit variance, one can draw a sample of size N from this distribution and form the sum of squares

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.