Topics
11. Bivariate Data
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United States of America
Grade 8

11.05 Interpret data with two-way tables

Lesson
Practice

Introduction

We have learned how to construct and interpret scatter plots for bivariate data. Using scatter plots and lines of best fit, we learned how to investigate patterns of association between two quantities.

We'll now try to organize bivariate, categorical data into a two-way table and use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Interpret data with two-way tables

Like Venn diagrams, two-way frequency tables are a visual way of representing information.

Two-way tables allow us to display and examine the relationship between two sets of categorical data. The categories are labeled at the top and the left side of the table, and the frequency of the different characteristics appear in the interior of the table. Often the totals of each row and column are also included.

The following table gives information on the treatment plans prescribed to cats with feline diabetes. Depending on the severity of the case, cats are either prescribed a special diet or insulin injections.

Special DietInsulin InjectionsTotal
Venetian Pet Clinic141428
Mohawk Animal Clinic11920
Total252348

It's called a two-way table because we can read information from it in two directions. Here we have information about the two categories clinic and treatment. If read across each row, we can tell how many cats in each clinic surveyed are treated by a special diet or insulin injections. If we read down each column, we can tell how many of the cats on special diet or on inuslin injections surveyed were in Venetian Pet Clinic and how many were in Mohawk Animal Clinic.

Notice the last column contains the totals for each row and the last row contains the totals for each column. They both add up to 48 which is the total number of cats surveyed.

When a particular row and column overlap, it indicates how many cats satisfy both categories. For example, there were 14 surveyed cats treated with insulin injections at the Venetian Pet Clinic.

Knowing there are 14 cats treated with insulin injections in Venetian Pet Clinic, we can find the percentage of cats in Venetian clinic injected with insulin by dividing 14 by the total number of cats treated in Venetian Clinic 28. We get \dfrac{14}{48}\times 100\% = 50\%. So we can conclude that \dfrac{1}{2} or 50\% of the pets surveyed in Venetian Pet CLinic were injected with insulin.

Examples

Example 1

Thirty six students were asked whether or not they were allergic to nuts and dairy. The two-way table is provided below:

Allergic to nutsNot allergic to nuts
Allergic to dairy106
Not allergic to dairy614
a

How many students are allergic to nuts?

Worked Solution
Create a strategy

Read across each row or column to find the category being asked.

Apply the idea

We want to find the number of students allergic to nuts.

Let's look at the first column.

We know that there are 10 students that are allergic to nuts and dairy.

We also know that there are 6 students that are allergic to nuts and not dairy.

How can we use these two values to find the total number of students that are allergic to nuts?

Number of students =10+6

Evaluate the addition.

Number of students =16

b

What percent of the students who are allergic to nuts are allergic to dairy?

Worked Solution
Create a strategy

We need the fraction of students allergic to dairy out of total number of students allergic to nuts written as a decimal.

Apply the idea
\displaystyle \% \text{ students allergic to dairy that are allergic to nuts}\displaystyle =\displaystyle \dfrac{\text{No.of students allergic to dairy}}{\text{Total no. of students allergic to nuts}}
\displaystyle =\displaystyle \dfrac{10}{16} \times 100\%
\displaystyle =\displaystyle 62.5\%
Reflect and check

This type of table allows us to examine if relationships exist between the categories. For example, for the given statistics above can you use supporting calculations to show if being allergic to dairy is dependent on being allergic to nuts?

c

What percent of students who are not allergic to dairy are not allergic to nuts?

Worked Solution
Create a strategy

We need the fraction of students not allergic to nuts out of total number of students who are not allergic to dairy written as a decimal.

Apply the idea

Looking at the second row:

Total number of students who are not allergic to dairy is 6+14=20

The number of students who are not allergic to nuts that are not allergic to dairy =14

\displaystyle \% \text{ students not allergic to nuts}\displaystyle =\displaystyle \dfrac{\text{No.of students not allergic to nuts}}{\text{Total no. of students not allergic to dairy}}
\displaystyle =\displaystyle \dfrac{14}{20} \times 100\%
\displaystyle =\displaystyle 70\%
Reflect and check

Of the students who are allergic to nuts, 62.5\% are allergic to dairy. Of the students who are not allergic to dairy, 70\% are not allergic to nuts. Is there a positive or negative correlation between having allergy to nuts and dairy?

Example 2

A total of 150 tennis players were asked whether they would support equal prize money for the women’s and men’s draw.

Out of 72 males, 37 are supporters. There are 12 female tennis players who are not supporters but 66 are females supporters.

Find the missing value in the table.

How many more players are there in support of equal prize money than those against it?

Find the percentage of the male tennis players who support equal prize money. Round your answer to one decimal place.

a

Complete the folowing table.

SupportDo not support
Males
Females
Worked Solution
Create a strategy

Use the information given to place in each category in the table.

Apply the idea
SupportDo not support
Males37
Females6612

To find the number of male who are not supporters, subtract the number of male supporters from the number of male tennis players.

\displaystyle \text{Male supporters}\displaystyle =\displaystyle 72-37Total male tennis players - male supporters
\displaystyle =\displaystyle 35Evaluate.
SupportDo not support
Males3735
Females6612
b

What percent of the tennis players are supporters.

Worked Solution
Create a strategy

We need to find the fraction of total number of supporters out of total number of tennis players in decimals.

Apply the idea

Looking at the first column, we can add the number of male supporters and female supporters. We'll use this to get the percentage of tennis players who are supporters.

\displaystyle \text{Total number of supporters}\displaystyle =\displaystyle 37+66Male supporters plus female supporters
\displaystyle =\displaystyle 103Evaluate
\displaystyle \% \text{ supporters}\displaystyle =\displaystyle \dfrac{\text{Total number of supporters}}{\text{Total number of tennnis players}}
\displaystyle =\displaystyle \dfrac{103}{150}\times 100\%
\displaystyle \approx\displaystyle 68.7\%Evaluate

Of the 150 tennis players, approximately 68.7\% are supporters.

Idea summary

A two-way table provides a way to organize data between two categorical variables.

We can use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Outcomes

8.SP.A.4

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

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