Two-Way Tables & Frequencies — I can create a two-way table from data and compute joint, marginal, and conditional relative frequencies.

Humans have been organizing information into tables for thousands of years, but the idea of using a table to compare two categorical variables at the same time is surprisingly modern. Before two-way tables existed, researchers struggled to show how two different categories—like gender and voting preference, or age group and product choice—related to each other. The development of these tables gave scientists, doctors, and social researchers a powerful tool for spotting patterns in data without complicated math.

The central question a two-way table answers is straightforward: How are two categorical variables distributed together, and is there a relationship between them? By the end of this lesson, you will be able to build a two-way table from raw data and extract three types of relative frequencies that reveal different layers of the story your data is telling.

Before diving into calculations, you need a solid grasp of the vocabulary. A two-way table (also called a contingency table) displays the counts for every combination of two categorical variables. One variable defines the rows and the other defines the columns. The cells inside the table hold the counts, and the edges hold the totals.

The diagram below shows a two-way table for a survey of 200 students who were asked about their preferred study method (alone or in a group) and whether they earned an A on the last exam. Study the layout carefully—notice where the joint frequencies, marginal totals, and grand total are placed.

Notice that each row total equals the sum of its interior cells: 40 + 60 = 100 for students who study alone, and 50 + 50 = 100 for students who study in a group. Similarly, each column total is the sum down that column: 40 + 50 = 90 students earned an A, and 60 + 50 = 110 did not. The grand total—200—can be found by adding either the row totals (100 + 100) or the column totals (90 + 110). This internal consistency is your best error-check when building tables.

Once you have a two-way table of raw counts, you can convert those counts into three types of relative frequencies. Each one divides a count by a different total, and each answers a different question.

Let's see all three frequency types side by side using the study-method data. The diagram below transforms the same raw data into a joint relative frequency table, then shows how conditional relative frequencies are computed by row. Comparing these three views of the same data is the key to mastering two-way tables.

In the joint relative frequency table (②), notice that all four interior cells plus the four cells add up to 1.00 (or equivalently, both marginal row totals sum to 1.00, and both marginal column totals sum to 1.00). In the conditional table (③), each row sums to 1.00 because we are asking about the distribution within that specific row. If you computed conditional frequencies by column instead, each column would sum to 1.00.

A school surveyed 150 students about whether they play a sport (Yes or No) and whether they have a part-time job (Yes or No). The raw data shows: 45 play a sport and have a job, 30 play a sport but have no job, 25 do not play a sport but have a job, and 50 neither play a sport nor have a job. Let's construct the two-way table and compute all three types of relative frequencies.

Each type of relative frequency answers a different question, and choosing the wrong one can lead to misleading conclusions. The table below summarizes when to use each type and what pitfalls to watch for.

Two-way tables are your entry point into a much larger world of statistical analysis. Once you are comfortable computing conditional relative frequencies, you are only a short step away from formal hypothesis testing and probability theory. The table below shows how the concepts you learned today connect to what comes next.

In AP Statistics and college-level courses, you will use the chi-square test to determine whether the differences you observe in conditional relative frequencies are large enough to be considered statistically significant, or whether they could have occurred just by chance. You will also encounter Simpson's Paradox, a fascinating phenomenon where an association that appears in several groups can reverse or disappear when the groups are combined. Both of these ideas rely heavily on the two-way table skills you are building right now.

Use the following two-way table for Problems 1–4. A school asked 250 students whether they prefer online or in-person classes and whether they are in STEM or non-STEM programs.