Name Class Date

8.1 Two-Way Frequency Tables

Essential Question: How can categorical data for two categories be summarized?

Resource
Locker
Explore Categorical Data and Frequencies
Data that can be expressed with numerical measurements are quantitative data. In this lesson you will examine
qualitative data, or categorical data, which cannot be expressed using numbers. Data describing animal type,
model of car, or favorite song are examples of categorical data.

 Circle the categorical data variable. Justify your choice.

temperature weight height color

B Identify whether the given data is categorical or quantitative.

large, medium, small

2 2
120 ft , 130 ft 2, 140 ft

 A frequency table shows how often each item occurs in a set of categorical data. Use the
categorical data listed on the left to complete the frequency table.

Ways Students Get to School Way Frequency

bus car walk car car car bus bus 8
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walk walk walk bus bus car

bus bus walk bus car bus car car

Reflect

1. How did you determine the numbers for each category in the frequency column?

2. What must be true about the sum of the frequencies in a frequency table?

Module 8 347 Lesson 1

 Explain 1 Constructing Two-Way Frequency Tables
If a data set has two categorical variables, you can list the frequencies of the paired values
in a two-way frequency table.

Example 1 Complete the two-way frequency table.

 A high school’s administration asked 100 randomly selected students in the 9th and 10th
grades about what fruit they like best. Complete the table.

Preferred Fruit
Grade Apple Orange Banana Total
9th 19 12 23
10th 22 9 15
Total
Row totals: Column totals: Grand total:
9th: 19 + 12 + 23 = 54 Apple: 19 + 22 = 41 Sum of row totals: 54 + 46 = 100
10th: 22 + 9 + 15 = 46 Orange: 12 + 9 = 21 Sum of column totals: 41 + 21 + 38 = 100

Banana: 23 + 15 = 38 Both sums should equal the grand total.

Preferred Fruit
Grade Apple Orange Banana Total
9th 19 12 23 54
10th 22 9 15 46
Total 41 21 38 100

B Jenna asked some randomly selected students whether they preferred dogs, cats, or other
pets. She also recorded the gender of each student. The results are shown in the two-way
frequency table below. Each entry is the frequency of students who prefer a certain pet and
are a certain gender. For instance, 8 girls prefer dogs as pets. Complete the table.

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Preferred Pet
Gender Dog Cat Other Total

Girl 8 7 1

Boy 10 5 9

Total

Row totals: Column totals: Grand total:

Girl: 8 + 7 + 1 = Dog: 8 + 10 = Sum of row totals: 16 + =

Boy: 10 + 5 + 9 = Cat: 7 + 5 = Sum of column totals: 18 + + =

Other: 1 + 9 = Both sums should equal the grand total.

Module 8 348 Lesson 1

 Reflect

3. Look at the totals for each row. Was Jenna’s survey evenly distributed among boys and girls? Explain.

4. Look at the totals for each column. Which pet is preferred by the most students? Justify your answer.

Your Turn

Complete the two-way frequency table.

5. Antonio surveyed 60 of his classmates about their participation in school activities and whether they have a
part-time job. The results are shown in the two-way frequency table below. Complete the table.

Activities
Job Clubs Only Sports Only Both Neither Total
Yes 12 13 16 4
No 3 5 5 2
Total

6. Jen surveyed 100 students about whether they like baseball or basketball. Complete the table.

Like Basketball
Like Baseball Yes No Total
Yes 61 13
No 16 10
Total

Explain 2 Reading Two-Way Frequency Tables

You can extract information about paired categorical variables by reading a two-way frequency table.
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Example 2 Read and complete the two-way frequency table.

 Suppose you are given the circled information in the table and instructed to complete
the table.

Eat Cereal for Breakfast

Gender Yes No Total
Girl 42 12 54
Boy 36 10 46
Total 78 22 100

Find the total number of boys by subtracting: 100 - 54 = 46

Find the number of boys who do eat cereal by subtracting: 46 - 10 = 36
Add to find the total number of students who eat cereal and the total number of students
who do not eat cereal.
Module 8 349 Lesson 1
B One hundred students were surveyed about which beverage they chose at lunch. Some of
the results are shown in the two-way frequency table below. Complete the table.

Lunch Beverage
Gender Juice Milk Water Total

Girl 10 17

Boy 15 24 21 60

Total

Find the total number of girls by subtracting: 100 - 60 =

So, the total number of girls is . The number of girls who do not choose milk is + = .

Find the number of girls who chose milk by subtracting: - =

Reflect

7. Which lunch beverage is the least preferred? How do you know?

Your Turn

Read and complete the two-way frequency table.

8. 100 students were asked what fruit they chose at lunch. The two-way frequency table shows some of the
results of the survey. Complete the table.

Lunch Fruit
Gender Apple Pear Banana Total
Girl 17 11 49

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Boy 10 16
Total

9. 200 high school teachers were asked whether they prefer to use the chalkboard or projector in class.
The two-way frequency table shows some of the results of the survey. Complete the table.

Preferred Teaching Aid

Gender Chalkboard Projector Total
Female 56 99
Male 44
Total 87 113 200

Module 8 350 Lesson 1

 Elaborate
10. You are making a two-way frequency table of 5 fruit preferences among a survey sample of girls and boys.
What are the dimensions of the table you would make? How many entries would you need to fill the table
with frequencies and totals?

11. A 3 categories-by-3 categories two-way frequency table has a row with 2 numbers, and no row or column
totals. Can you fill the row?

12. Essential Question Check-In How can you summarize categorical data for 2 categories?

Evaluate: Homework and Practice

• Online Homework
1. Identify whether the given data is categorical or quantitative. • Hints and Help
• Extra Practice
gold medal, silver medal, bronze medal
100 m, 200 m, 400 m
2. A theater company asked its members to bring in canned food for a food drive.
Use the categorical data to complete the frequency table.

Cans Donated to Food Drive Cans Frequency

peas corn peas soup corn
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soup
corn soup soup corn peas
peas corn soup peas corn
peas
peas corn peas corn soup
corn peas soup corn corn

Complete the two-way frequency table.

3. James surveyed some of his classmates about what vegetable they like best.
Complete the table.

Preferred Vegetable
Grade Carrots Green Beans Celery Total
9th 30 15 24

10th 32 9 20

Total

Module 8 351 Lesson 1

4. A high school’s extracurricular committee surveyed a randomly selected group of
students about whether they like tennis and soccer. Complete the table.

Like Tennis
Like Soccer Yes No Total
Yes 37 20

No 16 15

Total

5. After a school field trip, Ben surveyed some students about which animals they liked
from the zoo. Complete the table.

Preferred Animal at a Zoo

Grade Lion Zebra Monkey Total
11th 9 15 14

12th 4 17 15

Total

6. Jill asked some randomly selected students whether they preferred blue, green, or
other colors. She also recorded the gender of each student. The results are shown in
the two-way frequency table below. Complete the table.

Preferred Color
Gender Green Blue Other Total
Girl 15 3 10

Boy 3 16 6

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Total

7. Kevin surveyed some students about whether they preferred soccer, baseball, or
another sport. He also recorded their gender. Complete the table.

Preferred Sport
Gender Soccer Baseball Other Total
Girl 33 7 10

Boy 15 27 7

Total

Module 8 352 Lesson 1

 8. A school surveyed a group of students about whether they like backgammon and
chess. They will use this data to determine whether there is enough interest for the
school to compete in these games. Complete the table.

Like Backgammon
Like Chess Yes No Total
Yes 10 61

No 5 3

Total

9. Hugo surveyed some 9th and 10th graders in regard to whether they preferred math,
English, or another subject. The results of the survey are in the following table.
Complete the table.

Preferred Subject
Grade Math English Other Total
9th 40 35 20

10th 41 32 17

Total

10. Luis surveyed some middle school and high school students
about the type of music they prefer. Complete the table.
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Preferred Music
School Level Country Pop Other Total
Middle School 18 13 23

High School 7 32 15

Total

11. Natalie surveyed some teenagers and adults on whether they prefer standard cars,
vans, or convertibles. Her results are in the following table. Complete the table.
Images/Shutterstock

Preferred Car Type

Age Standard Van Convertible Total
Adults 10 25 9

Teenagers 11 7 24

Total

Module 8 353 Lesson 1

12. Eli surveyed some teenagers and adults on whether they prefer apples, oranges, or
bananas. His results are in the following table. Complete the table.

Preferred Fruit
Age Apple Orange Banana Total
Adults 22 12 10

Teenagers 24 9 9

Total

200 students were asked to name their favorite science class. The results are shown
in the two-way frequency table. Use the table for the following questions.

Favorite Science Class

Gender Biology Chemistry Physics Total
Girl 42 39 23 104

Boy 45 32

Total
13. How many boys were surveyed? Explain how you found your answer.

14. Complete the table. How many more girls than boys chose biology as their favorite
science class? Explain how you found your answer.

The results of a survey of 150 students about whether they

own an electronic tablet or a laptop are shown in the two-way
frequency table.

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Device
Gender Electronic tablet Laptop Both Neither Total
Girl 15 54 9 88

Boy 35 8 5

Total
15. Complete the table. Do the surveyed students own more laptops or more
electronic tablets?

16. Which group had more people answer the survey, boys or students who own an
electronic tablet only? Explain.

Module 8 354 Lesson 1

 17. The table shows the results of a survey about students’ preferred frozen yogurt flavor.
Complete the table, and state the flavors that students preferred the most and the least.

Preferred Flavor
Gender Vanilla Mint Strawberry Total
Girl 15 18 45
Boy 17 25
Total 100

18. Teresa surveyed 100 students about whether they like pop music or country music. Out
of the 100 students surveyed, 42 like only pop, 34 like only country, 15 like both pop and
country, and 9 do not like either pop or country. Complete the two-way frequency table.

Like Pop
Like Country Yes No Total
Yes
No
Total
19. Forty students in a class at an international high school were surveyed about which
non-English language they can speak. Complete the table.

Foreign Language
Gender Chinese Spanish French Total
Girl 7 8
Boy 6 7 18
Total

Luis surveyed 100 students about whether they like soccer. The number of girls and
the number of boys completing the survey are equal.

20. Complete the table.

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Likes Soccer
Gender Yes No Total
Girl 20
Boy 35
Total 100
21. Twice as many girls like soccer as the number that like tennis. The same number of
students like soccer as like tennis. Construct a table containing the tennis data.

Likes Tennis

Module 8 355 Lesson 1

22. A group of 200 high school students were asked about their use of email and text
messages. The results are shown in the two-way frequency table. Complete the table.

Text Messages
Email Yes No Total
Yes 72 90

No 45

Total

23. Circle the letter of each data set that is categorical. Select all that apply.
A. 75°, 79°, 77°, 85°

B. apples, oranges, pears

C. male, female

D. blue, green, red

E. 2 feet, 5 feet, 12 feet

F. classical music, country music

G. 1 centimeter, 3 centimeters, 9 centimeters

24. Explain the Error Find the mistake in completing the two-way frequency table for
a survey involving 50 students. Then complete the table correctly.

Favorite Foreign Language Class

Gender Russian German Italian Total
Girl 8 8 8 24

Boy 42 9 7 58

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Total 50

Correct table:

Favorite Foreign Language Class

Gender Russian German Italian Total
Girl 8 8 8 24

Boy 9 7

Total

Module 8 356 Lesson 1

 H.O.T. Focus on Higher Order Thinking

25. Justify Reasoning Charles surveyed 100 boys about their favorite color. Of the
100 boys surveyed, 44 preferred blue, 25 preferred green, and 31 preferred red.
a. Explain why it is not possible to make a two-way frequency table from the given
data.

b. Suppose Charles also surveyed some girls. Of the girls surveyed, 30 preferred blue
and 43 preferred green. Can Charles make a two-way frequency table now? Can
he complete it?

26. Persevere in Problem Solving Shown are two different tables about a survey
involving students. Each survey had a few questions about musical preferences.
All students answered all questions. Complete the tables. What type of music do
the students prefer?

Likes Classical Music

Gender Yes No Total
Girl 21

Boy 22

Total 100

Likes Blues Music

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Gender Yes No Total

Girl 15 49

Boy 15

Total

Module 8 357 Lesson 1

Lesson Performance Task
Two hundred students were asked about their favorite sport. Of the 200 students surveyed,
98 were female. Some of the results are shown in the following two-way frequency table.

Favorite Sport
Gender Football Baseball Basketball Soccer Total
Female 36 12

Male 38 19

Total 64 36

a. Complete the table.

b. Which sport is the most popular among the students? Which is the least popular?
Explain.

c. Which sport is most popular among the females? Which sport is most popular
among the males? Explain.

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Module 8 358 Lesson 1