A statistical test used to determine if observed data significantly differs from expected data, especially with categorical variables.

The actual frequencies you see in your data.

The frequencies you'd expect if there was no relationship between variables.

A table that organizes categorical data, making it easier to calculate expected counts and perform the chi-square test.

A table showing the distribution of categorical data, used to calculate expected counts and perform the chi-square test.

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

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.