AP Statistics · Topic 2.1 · Unit 2
AP Stats 2.1: Two-Way Tables
By Jude Wallis · Published
Topic 2.1 in the Fall 2026 AP Statistics course covers two-way tables (also called contingency tables) and graphs such as side-by-side and segmented bar charts, and mosaic plots. You use them to display two categorical variables together and judge whether the variables are associated.
AP Statistics: Unit 2 (topics 2.1). In the Fall 2026 AP Statistics course, tabular and graphical representations for two categorical variables are Unit 2 Topic 2.1, aligned to skills 4.A and 4.B.
What topic 2.1 covers
Topic 2.1 opens Unit 2 by looking at two categorical variables at once. The learning objective is to compare tabular and graphical representations of the relationship between those two variables, and then to justify a claim from what you see.
A categorical variable records a group or label rather than a number, such as grade level or whether someone owns a car. When you record two such variables for the same individuals, a two-way table and a few standard graphs let you compare the groups side by side.
Reading a two-way table
A two-way table, also called a contingency table, summarizes two categorical variables in a grid. One variable labels the rows and the other labels the columns, and each inner cell holds the count of individuals in that row-and-column combination.
The entries can be frequencies, meaning raw counts, or relative frequencies, meaning proportions of a total. The right-hand and bottom margins hold the row totals and column totals, and the single number in the corner is the grand total for everyone in the table. Turning those counts into joint, marginal, and conditional relative frequencies is the work of topic 2.2, later in Unit 2.
Graphs for two categorical variables
The CED names three graphs for showing two categorical variables together: side-by-side bar charts, segmented bar charts, and mosaic plots. In each one, the frequency or relative frequency of each level of one variable is displayed for every level of the other variable.
A side-by-side bar chart draws separate bars for each group next to one another. A segmented (stacked) bar chart stacks the levels of one variable inside a single bar per group, which makes proportions easy to compare when each bar is scaled to 100%. A mosaic plot does the same with bar widths that also reflect group size. These are close cousins of the plots in histogram vs bar graph.
Deciding whether the variables are associated
The payoff of these displays is deciding whether the two variables are associated. Two categorical variables are associated when the distribution of one variable changes across the levels of the other.
To check, compare the conditional distributions across groups: if the proportions in each category stay about the same from group to group, there is little evidence of association, and if they shift noticeably, that is evidence the variables are related. You then justify the claim by pointing to the specific numbers or bars, which is the skill later measured formally by a chi-square test.
Comparing car ownership across two grades
A survey of 200 students records grade (junior or senior) and whether the student owns a car. Among 100 juniors, 30 own a car; among 100 seniors, 55 own a car. Do the two variables appear associated?
Find the proportion of juniors who own a car: , or 30%.
Find the proportion of seniors who own a car: , or 55%.
Compare the two conditional proportions: , a gap of 25 percentage points.
A segmented bar chart scaled to 100% would show the car-owning segment much taller for seniors than for juniors.
Seniors own cars at a higher rate than juniors (55% versus 30%), a 25 percentage point gap that is evidence the two variables are associated.
Frequently asked questions
What is a two-way table in AP Statistics?
A two-way table, or contingency table, is a grid that summarizes two categorical variables at once. One variable labels the rows and the other labels the columns, and each cell holds the count (or proportion) of individuals in that combination. Row totals, column totals, and a grand total sit in the margins.
How do you tell if two categorical variables are associated?
Compare the conditional distributions across groups. If the proportion in each category stays roughly the same from one group to the next, there is little evidence of association. If the proportions shift noticeably, the variables appear associated. Segmented bar charts scaled to 100% make this comparison easy to see.