AP Statistics · Lesson 1 of 11 · Unit 2

Two categorical variables: two-way tables, graphs and relative frequencies

7 min read · about 1 h 30 min with practicePreview

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Analyzing two categorical variables bridges exploratory data analysis, conditional probability, and chi-square inference. On the AP Statistics exam, you must calculate specific relative frequencies from two-way tables, interpret segmented bar charts and mosaic plots, and use conditional proportions to justify whether two variables are associated.

By the end you’ll be able to

Compare side-by-side bar charts, segmented bar charts and mosaic plots to describe the relationship between two categorical variables.
Calculate joint, marginal and conditional relative frequencies from a two-way table.
Compare conditional relative frequencies across categories to decide whether two categorical variables appear to be associated.
Justify a claim about two categorical variables using tables, graphs or summary statistics in context.

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Core ideas (continued)
Worked examples
Exam technique
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What the exam asks

Section I (Multiple choice): Expect discrete questions (and possibly parts of the 3-question set on probability) that ask you to distinguish between joint, marginal, and conditional relative frequencies from a two-way table, or to match a table to its segmented bar chart or mosaic plot.
Section II (Free response): Can appear on Question 2 (Practices 3 and 4) or Question 4, where you may be asked to calculate proportions, show the fraction setup, and justify whether an association exists by comparing conditional relative frequencies in context.

Core ideas

Two-way tables and relative frequencies

A two-way table (or contingency table) displays counts for two categorical variables recorded from the same observational units. From a table with grand total $N$ , you can compute three types of relative frequencies (proportions or percentages):

Type	Definition	Denominator	Trigger words in the prompt
Joint relative frequency	Proportion of all individuals falling into a specific combination of both categories (an interior cell).	Grand total ( $N$ )	“and”, “both”, “of all surveyed... are [A] and [B]”
Marginal relative frequency	Proportion of all individuals falling into a single category of one variable.	Grand total ( $N$ )	“of all surveyed... are [A]” (ignores the second variable)
Conditional relative frequency	Proportion of individuals in a specific category of one variable who fall into a category of the other variable.	Row total or Column total	“of those who...”, “given that...”, “among the [A] group, what proportion are [B]”

Graphical displays for two categorical variables

To compare the distribution of a response variable across categories of an explanatory variable, we use three main graphs:

Side-by-side bar chart: Places bars for one variable’s categories next to each other for each category of the second variable. Best when vertical heights represent conditional relative frequencies (not raw counts, which mislead when group sizes differ).
Segmented bar chart: Stacks the conditional relative frequencies of one variable inside a single bar for each category of the other variable. Every bar has a total height of $100\%$ (or $1.0$ ), making it easy to compare segment heights across bars.
Mosaic plot: A modified segmented bar chart where the width of each bar is proportional to the marginal relative frequency (group size) of the explanatory variable on the horizontal axis. Because both height and width carry meaning, the area of each rectangle is proportional to the joint relative frequency of that cell.