Two-way tables
Also known as: contingency tables, cross tabulations
A two-way table organizes data by two categorical variables at once, with one variable defining the rows and the other the columns. Each cell shows how many observations share that row-column combination.
A two-way table (also called a contingency table) displays counts for two categorical variables simultaneously. Rows represent the categories of one variable and columns the categories of the other. For example, a survey of 100 students might use rows for grade level (junior, senior) and columns for transportation (bus, car, walk); the cell where "senior" meets "car" counts seniors who drive.
Two-way tables distinguish two kinds of frequencies. Joint frequencies are the interior cell counts — observations matching both categories. Marginal frequencies are the row and column totals in the table's margins, which describe one variable at a time. The grand total in the corner is the overall sample size, and every joint frequency divided by the grand total gives a joint relative frequency.
These tables are the natural setting for probability questions. The probability a randomly chosen student is a senior who drives is that cell over the grand total; a conditional probability like "given the student is a senior, what's the chance they drive?" divides the cell by the senior row total instead. Comparing conditional distributions across rows also reveals whether the two variables appear associated or independent.
Two-way tables show up regularly on quantitative exam sections. The GRE tests them alongside Venn diagrams for organizing overlapping categories, and the Praxis Core math test uses them for computing probabilities — so practice reading joint, marginal, and conditional values quickly.
Key takeaways
- A two-way table cross-classifies data by two categorical variables, one on rows and one on columns.
- Interior cells hold joint frequencies; row and column totals are marginal frequencies.
- Dividing a cell by the grand total gives a joint probability; dividing by a row or column total gives a conditional probability.
- Comparing conditional distributions reveals whether two variables are associated.
- The GRE and Praxis Core math exams test reading and computing probabilities from two-way tables.