Two-Way Tables and Relative Frequencies

Objective: Divide a cell by the grand total for a joint event.

Prerequisite skills: counts, fractions, and table totals. A two-way table cross-classifies observations by two categorical variables. A joint count belongs to one row and one column. Its whole-sample relative frequency is that cell divided by the grand total. Common error: dividing by a row total when the question refers to the whole sample. Restrictions and interpretation: Relative frequency uses the population specified in the question: whole table, row, or column.

Objective: Use the total of the condition group as the denominator.

A phrase such as among students restricts the reference population to the student row or column. Divide the desired overlap count by that group total, not by the full sample. Common error: using the grand total after the question restricts the group. Restrictions and interpretation: Relative frequency uses the population specified in the question: whole table, row, or column.

Objective: Use row and column totals to reconstruct a table.

A row total equals the sum of its cells, and a column total does the same across rows. Subtract known cells from a compatible marginal total. Verify all totals remain consistent and nonnegative. Common error: subtracting a cell from an unrelated row or column total. Restrictions and interpretation: Relative frequency uses the population specified in the question: whole table, row, or column. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.