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

$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$ where O is observed and E is expected.

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

Independence: One sample, two categorical variables, exploring relationships within a single group. | Homogeneity: Two or more samples, one categorical variable, comparing distributions across groups.

Independence: Requires a simple random sample (SRS). | Homogeneity: Requires a stratified random sample OR randomly assigned treatments (for experiments).

A test used to determine if there is a significant association between two categorical variables within a single population.

A test used to determine if the distribution of a categorical variable is the same across two or more populations or treatments.

The counts we would expect in each cell of a contingency table if the null hypothesis were true. Calculated as (Row Total * Column Total) / Grand Total.

There is no association between two categorical variables (they are independent).

There is no difference in the distribution of a categorical variable across populations/treatments.

There is an association between two categorical variables (they are dependent).

There is a difference in the distribution of a categorical variable across populations/treatments.