Two-Way Tables
Construct and read two-way tables that sort the same people by two categories at once (e.g., year group against preferred sport); use row and column totals and relative frequencies to judge whether the two categories are associated
What a learner can do afterwards
- Build a two-way table from raw data on two categories collected from the same people
- Read off joint and total frequencies (e.g., how many students both walk to school and pack their lunch)
- Compare row or column relative frequencies to say whether two things seem linked (e.g., do pet owners do more sport?)
1 · Read
You already know how to read a table. A two-way table does one more job: it sorts the same group of people by two questions at once, not just one.
Ms. Ito asked 50 students two questions: do you have a pet, and do you play a sport? She put the answers in a two-way table.
| Plays sport | No sport | Total | |
|---|---|---|---|
| Has a pet | 18 | 6 | 24 |
| No pet | 10 | 16 | 26 |
| Total | 28 | 22 | 50 |
The box where a row meets a column is called a cell. The 'Has a pet' row meets the 'Plays sport' column at 18. That means 18 students both have a pet and play a sport.
The 'Total' row and column tell you about just one question at a time. At the bottom of the 'Plays sport' column, 28 students play a sport in total, no matter whether they have a pet. In the right-hand 'Total' column, 24 students have a pet, no matter what sport they play.
Row totals are not always equal (24 pet owners, 26 without). That is why you compare percentages, not raw counts, when you want to know if two categories are linked.
A two-way table sorts the same people by two questions at once; compare relative frequencies, not raw counts, to see if the two are linked.
More to read
Probability and relative frequency
Two-way frequency tables help you organize information about two things at once. Picture a table with rows for one category and columns for another category, like which class a student is in and whether they went to a concert.
Each box inside the table (not a total box) tells you how many people fit both categories at the same time. That is called a joint frequency.
The totals at the end of each row and each column tell you how many people are in that group altogether, no matter what the other category was.
Once you have counts, you can turn them into percentages. That is called relative frequency. It lets you compare groups fairly, even if the groups are different sizes.
For example, if one teacher's class has 30 students and 18 went to a concert, that is 60%. Another teacher's class might have 40 students with 20 going, which is only 50%. Even though 20 is a bigger number than 18, the second class actually had a lower rate of attending. Relative frequency is what lets you spot that.
Two-way relative frequency tables and associations
A relative frequency table is a two-way table filled with percentages instead of plain counts. It still has rows for one category and columns for another, but every number inside shows what share of the group falls into that box.
Say a school asks 200 students whether they like campus housing, split by whether they are first-year students or upperclassmen. A plain two-way table would show raw counts, like 40 first-years who like it and 60 who do not. A relative frequency table turns those into percentages of the row or column total, so you can compare groups of different sizes fairly.
The real power of a relative frequency table is spotting a relationship. If the percentage who like housing is close to the same for first-years and upperclassmen, the two categories are probably not linked. But if one group's percentage is a lot higher than the other's, that is a sign there might be an association between the two categories.
7.8 The Addition Rule for Probability
Think about everyone in your class. You could sort them in a bunch of different ways at once: what year they are in school, what their favorite subject is, whether they play a sport, and more.
When you pick two of those categories and organize people by both at the same time, you get a two-way table. One category goes across the rows, and the other goes across the columns. Every person in the group gets counted in exactly one box, based on their answer to both categories.
This kind of table is useful because it lets you see the whole picture at once. You do not just see how many students play a sport overall. You can see how many students in each year group play a sport, which lets you compare across years instead of just looking at one big total.
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2 · Watch
En anglaisTake it off screen
Where it sits
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.