Two-Way Frequency Tables

In this lesson you will: Read counts and totals out of a two-way frequency table; Compute joint, marginal and conditional relative frequencies, and tell which one a question wants; Use conditional relative frequencies to judge whether two categorical variables are associated.

Two questions asked of the same people

A two-way frequency table records the answers to two yes-or-no questions from every person in one survey. Two hundred students at a high school were asked whether they play a school sport and whether they hold a part-time job. Each student lands in exactly one of the four inner cells.

The numbers around the edge are the margins. They are not extra information; they are sums. The row margins add across, the column margins add down, and both reach the same grand total.

Three fractions you can make from one cell

Take the count 36, the students who both play a sport and hold a job. Three different questions all have that 36 on top, and the difference between them is entirely in the denominator.

A joint relative frequency divides by the grand total. A marginal relative frequency divides a row or column total by the grand total. A conditional relative frequency restricts attention to one row or one column first, and divides by that group's own total.

Conditional frequencies are how an association shows up

Ask the same conditional question of each row and compare the answers. Among the students who play a sport, 36 of 120, or 30%, have a job. Among the students who do not, 44 of 80, or 55%, do.

Those rates are far apart, so knowing whether a student plays a sport changes what you would guess about their job. That is what an association between two categorical variables means. If the two rates had come out near 40% each, the table would show no association worth mentioning.

Common mistake — Dividing by the grand total when the question said "of"

The question reads: Of the students who play a sport, what percent have a part-time job? A student writes (36)/(200)=18%.

The pull toward 200 is strong, because 200 is the number printed in the corner and it worked for the last question. But the phrase of the students who play a sport has already thrown away 80 of the 200 people. The survey has shrunk to 120, and 120 is the denominator.

The correct answer is (36)/(120)=30%. Both numbers are meaningful, and each answers a real question — but only one answers this question.

How to catch it: Underline the words immediately after of. That phrase names your group, and the group's total goes underneath. Then run the row check: the conditional relative frequencies across one row must add to 1. Here (36)/(120)+(84)/(120)=0.30+0.70=1, which confirms the denominator. The wrong version gives 0.18+0.42=0.60, and a row that fails to reach 1 is a denominator error every time.