Math Core

Not all paired data is made of numbers. You might ask each student two yes-or-no questions, like "Do you play an instrument?" and "Do you play a sport?" You can't make a scatter plot of yes and no answers, but a two-way table organizes them so you can look for an association.

Reading a two-way table

A school surveyed $100$ eighth graders with those two questions. Every student falls into exactly one of four groups.

	Plays a sport	No sport	Total
Plays an instrument	18	22	40
No instrument	42	18	60
Total	60	40	100

Each inside cell counts students in both categories. For example, $18$ students play an instrument and a sport.
The Total column adds across each row: $18 + 22 = 40$ students play an instrument.
The Total row adds down each column: $18 + 42 = 60$ students play a sport.
The bottom-right corner is the grand total: $100$ students.

Definition

Two-way table

A table that shows data about two categorical variables at once. One variable's categories label the rows and the other's label the columns. Each cell shows a frequency: how many individuals fall into that row and that column.

Filling in missing values

Because the rows and columns must add up, you can often find missing numbers.

Worked example: Completing a table

Some entries are missing from this table about $80$ students.

	Has a pet	No pet	Total
Walks to school	12	?	30
Rides to school	?	?	?
Total	44	?	80

Fill in the table.

Solution. Work one row or column at a time.

Walks, no pet: $30 - 12 = 18$ .
Rides, has a pet: $44 - 12 = 32$ .
Total with no pet: $80 - 44 = 36$ .
Rides, no pet: $36 - 18 = 18$ .
Total who ride: $32 + 18 = 50$ . Check: $30 + 50 = 80$ . ✓

	Has a pet	No pet	Total
Walks to school	12	18	30
Rides to school	32	18	50
Total	44	36	80

Relative frequencies

Raw counts can be misleading when the groups are different sizes. In the survey, $18$ instrument players play a sport and $42$ non-players do. But there are more non-players to begin with, so comparing $18$ with $42$ isn't fair.

Instead, compare relative frequencies: each count divided by its row total (or column total). This turns counts into fractions or percents of the group.

Row relative frequency

To compare the rows, divide each cell by its row's total.

\text{relative frequency} = \frac{\text{count in the cell}}{\text{row total}}

Each row of relative frequencies adds to $1$ , or $100\%$ .

Worked example: Looking for an association

Use the instrument-and-sport survey. Is there an association between playing an instrument and playing a sport?

Solution. Divide each cell by its row total.

	Plays a sport	No sport	Total
Plays an instrument	$\dfrac{18}{40} = 45\%$	$\dfrac{22}{40} = 55\%$	$100\%$
No instrument	$\dfrac{42}{60} = 70\%$	$\dfrac{18}{60} = 30\%$	$100\%$

Only $45\%$ of instrument players also play a sport, but $70\%$ of the other students do. The percents are quite different, so there is an association: in this school, students who don't play an instrument are more likely to play a sport.

If the percents in the two rows were about the same, there would be no association. Knowing whether a student plays an instrument wouldn't help you guess whether they play a sport.

Worked example: Column relative frequencies

You can also divide by column totals. Of the students who play a sport, what percent play an instrument?

Solution. Use the "Plays a sport" column, whose total is $60$ :

\frac{18}{60} = 0.3 = 30\%

So $30\%$ of the athletes play an instrument.

Common mistake

Read the question carefully to know which total to divide by. "Of the instrument players, what percent play a sport?" uses the row total: $\dfrac{18}{40} = 45\%$ . "Of the athletes, what percent play an instrument?" uses the column total: $\dfrac{18}{60} = 30\%$ . Same cell, different answer. The group named after "of the" gives the denominator.

Tip

Before answering, check your table: every row should add to its row total, every column to its column total, and both the Total row and the Total column should add to the grand total.

Practice

Problems 1 to 4 use this table from a survey of $150$ students.

	Likes rap	Likes country	Total
7th grade	45	25	70
8th grade	60	20	80
Total	105	45	150

Practice 1

How many 8th graders prefer country music?