Chi-square test

A chi-square test works with counts in categories rather than means, checking goodness of fit, independence, or homogeneity. It adds up the squared gaps between observed and expected counts, each scaled by the expected count, into one statistic. The formula is $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is an observed count and $E$ an expected count. For example, rolling a die 60 times and comparing each face's tally to the expected 10 tests whether the die is fair, and a large $\chi^2$ gives a small p-value.