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Two-Way Tables/Contingency Tables
Common Core: 8.SP.4 S.CP.4 S.ID.5

Two-Way Tables/Contingency Tables

Author: Sophia Tutorial

This lesson will explain two-way tables.

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What's Covered

In this tutorial, you're going to learn about two-way tables, also called contingency tables. Specifically you will focus on:

  1. Two-Way Tables


Two-way tables area way of showing the relationship between two categorical variables.

Term to Know

  • Two-Way Tables
  • A way of presenting data such that we can see the relationships between two categorical variables.

Suppose you had 335 students in different parts of the country and they were asked the question, if you had to pick one thing about school that's most important to you: getting good grades, being popular, or being good at sports? The distribution looked like this.

This means that 57 rural students said the grades were the most important thing. Six urban students said that being popular was the most important thing to them. We can see the relationship between school location and goal.

One of the things that we can look at with a two-way table, one of its most important features, is called the marginal distributions. They are called that because they're written in the margins. They are the row totals and column totals for the particular categories that you have.

This shows that there were 149 rural students in this study, whereas there were only 35 urban students in this study. It shows you that 168 students said that grades were the most popular thing, regardless of where they live. And 98 students said that being popular was the most important thing at school.

We also can add up all these cell values and obtain the grand total. This means there were 335 students in the study, which we knew at the beginning, based on the way the problem was posed. This allows us to answer some pretty interesting probability problems.

  1. What's the probability that a student says that grades are the most important thing? That would be 168 students out of 335 students. Because there were 168 students, regardless of where they live, that said that grades were the most important thing, out of the 335.
  2. What's the probability that an urban student says popular? Isolate your view to just the 35 urban students, and say that it's six out of those 35.
  3. What's the probability that someone who said sports was a rural student? Limit your view to just the 69 students who said sports, and see that 42 out of those 69 were the ones who were in the rural schools.


Two-way tables help you to understand the relationships between two different events, or two different categories. Use two-way tables to answer some pretty interesting probability questions. Oftentimes, you use those marginal distributions, the row totals or column totals, or even the grand total.

Good luck.

Source: This work adapted from Sophia Author Jonathan Osters.

Terms to Know
Two-Way Table/Contingency Table

A way of presenting data such that we can see the relationships between two categorical variables.