Chi-squared test

Summary

A chi-squared test is a hypothesis test whose test statistic has approximately a chi-squared distribution under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories. For observed counts $O_1, \dots, O_k$ and expected counts $E_1, \dots, E_k$ under $H_0$, Pearson's statistic is $$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}.$$ One rejects $H_0$ when $\chi^2$ is large.

Remarks

In the goodness-of-fit test, $H_0$ fixes the probabilities of the $k$ categories. For a large sample, $\chi^2$ has approximately the law $\chi^2_{k - 1}$ under $H_0$. In the test of independence for an $r \times c$ contingency table, $H_0$ states that the two categorical variables are independent. Then $E_{ij}$ is (row total $\times$ column total) / $n$, and the law is approximately $\chi^2_{(r-1)(c-1)}$. The approximation is poor when expected counts are small. A common rule asks for $E_i \ge 5$ in each cell. For small tables, Fisher's exact test is an alternative. Karl Pearson introduced the test in 1900.

Explanations

L1 human

The chi-squared test compares the counts that you observe with the counts that a hypothesis predicts. If the squared differences, scaled by the predictions, add up to a large number, the hypothesis does not fit the data.