The Chi-Square Distribution

Summary :

The chi-square distribution is used to test claims about categorical data and variances. This chapter describes the properties of the chi-square distribution and applies it to goodness-of-fit tests, tests of independence between two variables, tests of homogeneity across populations, and tests about a single population variance.

Facts About the Chi-Square Distribution

The chi-square distribution is a family of right-skewed curves that take only non-negative values, because the statistic is built from squared quantities. Its shape depends on a single degrees-of-freedom value, and as that value grows the curve becomes more symmetric and bell-like. The distribution is used to compare observed counts with the counts expected under a hypothesis, and to draw conclusions about categorical data and about the variance of a population.

Goodness-of-Fit and Independence Tests

A goodness-of-fit test checks whether observed frequencies across categories match a claimed distribution, such as whether lottery numbers occur equally often. A test of independence uses a contingency table to decide whether two categorical variables are related or independent. In each case the test compares observed counts with expected counts, sums the standardised squared differences into a chi-square statistic, and compares it with the distribution to obtain a p-value and a decision.

Homogeneity and Single-Variance Tests

A test of homogeneity checks whether two or more populations share the same distribution across categories, extending the same counting logic to several groups at once. The chi-square distribution is also used to test a claim about a single population variance or standard deviation. Across all these uses the pattern is the same: measure how far the data depart from what a hypothesis predicts, express that gap as a chi-square value, and judge whether it is larger than chance would explain.