Chi-square test

If N measurements y_i are compared to some model or theory predicting values g_i, and if the measurements are assumed normally distributed around g_i, uncorrelated and with variances , then the sum

follows the (chi-square) distribution with N degrees of freedom. The test compares s with the integral of the distribution; if the sum above is equal to the quantile of the distribution

then the probability of obtaining s or a larger value in the 'null hypothesis' (i.e. the y_i are drawn from a distribution described by the ) is given by .

Integral curves for the distribution exist in computer libraries or are tabulated in the literature. Note that the test may express little about the inherent assumptions; wrong hypotheses or measurements can, but need not cause large 's. The only statement to make about a measured s is the one above: `` is the probability of finding a as large as s or larger, in the null hypothesis.

Rudolf K. Bock, Oct 2000