AP Statistics · Topic 2.2 · Unit 2
AP Stats 2.2: Two-Way Table Summary Stats
By Jude Wallis · Published
Topic 2.2 in the Fall 2026 AP Statistics course turns two-way table counts into proportions. A joint relative frequency divides a cell by the grand total, a marginal divides a row or column total by the grand total, and a conditional divides a cell by its own row or column total.
AP Statistics: Unit 2 (topics 2.2). In the Fall 2026 AP Statistics course, summary statistics for two categorical variables are Unit 2 Topic 2.2, aligned to skills 3.B, 4.A, and 4.B.
What topic 2.2 covers
Topic 2.2 turns the counts in a two-way table into proportions you can compare. The objective is to calculate summary statistics from two-way tables, then compare and justify claims about whether the two categorical variables are associated.
There are exactly three kinds of relative frequency to keep straight: joint, marginal, and conditional. The only difference among them is what you divide by.
Joint, marginal, and conditional relative frequency
A joint relative frequency is a single cell frequency divided by the grand total for the whole table. It answers a question about both variables at once, such as the proportion of all students who are juniors and own a car.
A marginal relative frequency is a row total or a column total divided by the grand total. It describes just one variable while ignoring the other, such as the proportion of all students who own a car.
A conditional relative frequency restricts attention to one level, or category of interest, first. You divide a cell frequency by the total of its row (or the total of its column), which gives a proportion within that single group.
Comparing conditional distributions for association
Conditional relative frequencies are the ones that reveal association, because they let you compare the same category across different groups. If the conditional proportion of car owners is much higher among seniors than among juniors, that difference is evidence the two variables are associated.
Marginal and joint frequencies describe the overall table but hide the group-to-group comparison, so they cannot by themselves show association. When you justify a claim, quote the specific conditional proportions and the context, the way the reasoning behind a chi-square test does more formally.
Keeping the three straight
The three relative frequencies are easy to mix up because they draw on the same cells. A joint relative frequency always uses the grand total as its denominator, so every joint frequency in a table adds to 1. A marginal relative frequency uses a margin total over the grand total, so the row marginals add to 1 on their own and the column marginals add to 1 on their own.
A conditional relative frequency uses a single group's total as its denominator, so the conditional frequencies within one row (or within one column) add to 1. A fast way to identify which kind you have is to ask which set of proportions sums to 1. That check also flags an arithmetic slip before it reaches your final answer.
Three relative frequencies from one table
Of 200 students, 100 are juniors (30 own a car, 70 do not) and 100 are seniors (55 own a car, 45 do not). So 85 own a car and 115 do not overall. Find one joint, one marginal, and one conditional relative frequency.
Joint (junior and owns a car): divide the cell by the grand total, .
Marginal (owns a car, either grade): divide the column total by the grand total, .
Conditional (owns a car given senior): divide the cell by the senior row total, .
Conditional the other direction (senior given owns a car): divide by the car-owner total, .
Joint , marginal , and the conditional proportion of car owners among seniors is , higher than the among juniors, which points to association.
Frequently asked questions
What is the difference between joint, marginal, and conditional relative frequency?
All three are proportions from a two-way table; the difference is the denominator. Joint divides a cell by the grand total. Marginal divides a row or column total by the grand total. Conditional divides a cell by its own row total or column total, restricting attention to one group.
Which relative frequency shows association between two variables?
Conditional relative frequencies do. Because they compare the same category across different groups, a noticeable shift in the conditional proportions from group to group is evidence the two categorical variables are associated. Joint and marginal frequencies describe the overall table but hide that comparison.