The chi-square statistic (or -statistic) measures agreement between the observed and hypothetical frequencies. This statistic is computed from two entities: hypothetical probabilities of the values of a discrete random variable , and the observed frequencies of these values - the numbers of observations of each type. The chi-square statistic is the heart of the chi-square test .
The chi-square statistic is computed according to the following formula:
is the number of observed events of the th type;
is the hypothetical probability of the event of the th type.
For small samples, the sampling distribution of this statistic is not guaranteed to be the chi-square distribution. In such situations resampling is often used (see more on resampling in the online book Resampling: The New Statistics ).
There are other statistics that obey the chi-square distribution and, therefore, might be called "chi-square statistics" too. But the statistic described here is the most famous one - it is discussed even in introductory texts in statistics and widely used in practice.