Homogeneity: Compares across populations, looking for similar distributions. | Independence: Looks within a single population, testing for associations between variables.

Homogeneity: The distribution of the categorical variable is the same across all populations. | Independence: There is no association between the two categorical variables.

A grid showing the relationship between two categorical variables.

Compares the distribution of a categorical variable across two or more independent groups/populations.

Examines the relationship between two categorical variables within a single population.

The counts we expect to see if there's no association (independence) or no difference (homogeneity).

States that there is no association between the two categorical variables.

Expected Count = (Row Total * Column Total) / Table Total

$$\chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}$$

$$df = (number\ of\ rows - 1) * (number\ of\ columns - 1)$$