Test for homogeneity

Math

Definition

A chi-square test for homogeneity determines whether the distribution of a categorical variable is the same across two or more independent populations or groups. It uses a two-way table and compares observed counts to expected counts under the assumption that proportions are equal across groups.

How It Works

  1. State H₀: the distribution of the variable is the same across all populations.
  2. Create a two-way table of observed frequencies.
  3. Calculate expected counts: (row total × column total) / grand total.
  4. Compute the chi-square statistic: χ² = Σ (observed − expected)² / expected.
  5. Find the p-value using df = (rows − 1)(columns − 1) and make a conclusion.

Examples

  • Comparing whether the distribution of favorite music genres is the same for students at three different schools
  • Testing if the proportion of voters supporting a policy is the same across four states
  • Checking if customer satisfaction ratings are distributed the same way at multiple store locations
Key Fact

χ² = Σ (O − E)² / E, with df = (r − 1)(c − 1)

Related Terms

Test for independence Goodness of fit Hypothesis tests for proportions

Study This Concept

Practice test for homogeneity with free review games in these units:

AP Statistics Unit 8: Chi-Square Tests