Independence: One sample, tests for association between two categorical variables. | Homogeneity: Multiple samples, tests if the distribution of a categorical variable is the same across populations.

Observed Frequency: Actual counts from the sample data. | Expected Frequency: Counts expected if the null hypothesis is true (no association).

Rejecting H0: There is statistically significant evidence for the alternative hypothesis. | Failing to Reject H0: There is not enough evidence to support the alternative hypothesis.

A statistical test used to analyze categorical data to determine if there's a significant association between variables or if observed data fits an expected pattern.

There is no association between the variables (for independence) or the distributions are the same (for homogeneity).

There is an association between the variables (for independence) or the distributions are different (for homogeneity).

The actual count of data points falling into a specific category.

The count we would expect in a cell if the null hypothesis were true.

The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

$\chi^2 = \sum \frac{(O - E)^2}{E}$

$E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$

$df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$