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

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

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

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

Expected Frequency = Total Number of Observations * Hypothesized Proportion

It examines if a sample distribution matches a hypothesized distribution. It tests if your data "fits" a particular model.

It determines if two categorical variables are independent of each other. It assesses whether they are related or if the observed relationship is due to chance.

It compares distributions of a categorical variable across different populations or treatments to determine if the groups are similar or different.

All expected counts must be at least 5. This ensures the sampling distribution of the test statistic is approximately chi-square.

A low p-value means we have evidence to reject the null hypothesis, suggesting a significant association or difference.

Independence: Tests relationship between two variables in a single population. | Homogeneity: Compares distributions of a variable across multiple populations.

Observed: Actual counts in your sample data. | Expected: Counts predicted under the null hypothesis (no association).

Goodness of Fit: Tests if one sample matches a hypothesized distribution. | Independence: Tests if two categorical variables are related.

Chi-Square: Used for categorical variables with two or more categories. | Z-test: Used for comparing proportions of a single categorical variable with two categories.

Rejecting: There is enough evidence to support the alternative hypothesis. | Failing to reject: There is not enough evidence to support the alternative hypothesis.