Chi-square tests

A chi-square goodness-of-fit test compares observed category counts to expected counts derived from a hypothesized distribution. The test statistic is ${\chi }^2=\sum \frac{(O-E{)}^2}{E}$, where $O$ is an observed frequency and $E$ is its expected frequency under the null hypothesis. The degrees of freedom equal the number of categories minus one. Reject $H_0$ if the p-value is less than the significance level, meaning the observed distribution differs significantly from the expected distribution.