Name Class 10-1
Date Organizing and Displaying Data
Extension: Two-Way Frequency Tables
Essential question: How can categorical data be organized and analyzed?
In previous lessons, you worked with numerical data involving variables
such as age and height. In this lesson, you will analyze categorical data
that involve variables such as pet preference and gender. The frequency
of a data value is the number of times it occurs. A frequency table
shows the frequency of each data value.
Video Tutor
CC.9–12.S.ID.5
1
EXAMPLE
Creating a Relative Frequency Table
The frequency table below shows the results of a survey that Jenna took at her
school. She asked 40 randomly selected students whether they preferred dogs,
cats, or other pets. Convert this table to a relative frequency table that uses
decimals as well as one that uses percents.
Preferred Pet
Frequency
A
Dog
Cat
Other
Total
18
12
10
40
Divide the numbers in the frequency table by the total to obtain relative frequencies
as decimals. Record the results in the table below.
Preferred Pet
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Relative
Frequency
Dog
Cat
Other
Total
Other
Total
18
​ ___
  ​= 0.45
40
B Write the decimals as percents in the table below.
Preferred Pet
Dog
Relative
Frequency
45%
Cat
REFLECT
1a. How can you check that you have correctly converted frequencies to relative
frequencies?
1b. Explain why the number in the Total column of a relative frequency table is
always 1 or 100%.
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Lesson 1
In the previous example, the categorical variable was pet preference, and the variable
had three possible data values: dog, cat, and other. The frequency table listed the
frequency for each value of that single variable. If you have two categorical variables
whose values have been paired, you list the frequencies of the paired values in a
two-way frequency table.
2
CC.9–12.S.ID.5
EXAMPLE
Creating a Two-Way Frequency Table
For her survey, Jenna 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.
Preferred
Pet
Gender
Dog
Cat
Other
Girl
8
7
1
Boy
10
5
9
Total
Total
A Find the total for each gender by adding the frequencies in each row. Write the row
totals in the Total column.
B Find the total for each preferred pet by adding the frequencies in each column. Write
the column totals in the Total row.
C Find the grand total, which is the sum of the row totals as well as the sum of the
© Houghton Mifflin Harcourt Publishing Company
column totals. Write the grand total in the lower-right corner of the table (the
intersection of the Total column and the Total row).
REFLECT
2a. Where have you seen the numbers in the Total row before?
2b. In terms of Jenna’s survey, what does the grand total represent?
You can obtain the following relative frequencies from a two-way frequency table:
• A joint relative frequency is found by dividing a frequency that is not in the
Total row or the Total column by the grand total.
• A marginal relative frequency is found by dividing a row total or a column total
by the grand total.
Chapter 10
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Lesson 1
A two-way relative frequency table displays both joint relative frequencies and marginal
relative frequencies.
CC.9–12.S.ID.5
3
EXAMPLE
Creating a Two-Way Relative Frequency Table
Create a two-way relative frequency table for Jenna’s data.
A Divide each number in the two-way frequency table from the previous example by the
grand total. Write the quotients as decimals.
Preferred
Pet
Gender
Dog
Girl
___
​ 8  ​ = 0.2
40
Cat
Other
Total
Boy
Total
___
​ 18  ​= 0.45
40
___
​ 40  ​= 1
40
B Check by adding the joint relative frequencies in a row or column to see if the sum
© Houghton Mifflin Harcourt Publishing Company
equals the row or column’s marginal relative frequency.
0.2 +
+
=
Girl row:
Boy row:
Dog column:
Cat column:
+
=
Other column:
+
=
+
+
0.2 +
=
= 0.45
REFLECT
3a. A joint relative frequency in a two-way relative frequency table tells you what
portion of the entire data set falls into the intersection of a particular value of one
variable and a particular value of the other variable. For instance, the joint relative
frequency of students surveyed who are girls and prefer dogs as pets is 0.2, or 20%.
What is the joint relative frequency of students surveyed who are boys and prefer
cats as pets?
3b. A marginal relative frequency in a two-way relative frequency table tells you
what portion of the entire data set represents a particular value of just one of
the variables. For instance, the marginal relative frequency of students surveyed
who prefer dogs as pets is 0.45, or 45%. What is the marginal relative frequency of
students surveyed who are girls?
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Lesson 1
One other type of relative frequency that you can obtain from a two-way frequency table
is a conditional relative frequency. A conditional relative frequency is found by dividing
a frequency that is not in the Total row or the Total column by the frequency’s row total or
column total.
CC.9–12.S.ID.5
4
EXAMPLE
Calculating Conditional Relative Frequencies
From Jenna’s two-way frequency table you know that 16 students surveyed are
girls and 12 students surveyed prefer cats as pets. You also know that 7 students
surveyed are girls who prefer cats as pets. Use this information to find each
conditional relative frequency.
A Find the conditional relative frequency that a student surveyed prefers cats as pets,
given that the student is a girl.
Divide the number of girls who prefer cats as pets by the number of girls. Express your
answer as a decimal and as a percent.
B Find the conditional relative frequency that a student surveyed is a girl, given that the
student prefers cats as pets.
Divide the number of girls who prefer cats as pets by the number of students who
prefer cats as pets. Express your answer as a decimal and as a percent.
REFLECT
4a. When calculating a conditional relative frequency, why do you divide by a row total
or a column total and not by the grand total?
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4b. You can obtain conditional relative frequencies from a two-way relative frequency
table. For instance, in Jenna’s survey, the relative frequency of girls who prefer
cats as pets is 0.175, and the relative frequency of girls is 0.4. Find the conditional
relative frequency that a student surveyed prefers cats as pets, given that the
student is a girl.
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Lesson 1
CC.9–12.S.ID.5
5
EXAMPLE
Finding Possible Associations Between Variables
Jenna conducted her survey because she was interested in the question, “Does
gender influence what type of pet people prefer?” If there is no influence,
then the distribution of gender within each subgroup of pet preference should
roughly equal the distribution of gender within the whole group. Use the results
of Jenna’s survey to investigate possible influences of gender on pet preference.
A Identify the percent of all students surveyed who are girls:
B Determine each conditional relative frequency.
Of the 18 students who prefer dogs as pets, 8 are girls.
Percent who are girls, given a preference for dogs as pets:
Of the 12 students who prefer cats as pets, 7 are girls.
Percent who are girls, given a preference for cats as pets:
Of the 10 students who prefer other pets, 1 is a girl.
Percent who are girls, given a preference for other pets:
C Interpret the results by comparing each conditional relative frequency to the percent
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of all students surveyed who are girls.
The percent of girls among students who prefer dogs is fairly close to 40%, so gender
does not appear to influence preference for dogs.
The percent of girls among students who prefer cats is much greater than 40%. What
conclusion might you draw in this case?
The percent of girls among students who prefer other pets is much less than 40%. What
conclusion might you draw in this case?
REFLECT
5a. Suppose you analyzed the data by focusing on boys rather than girls.
How would the percent in Part A change? How would the percents in Part B
change? How would the conclusions in Part C change?
5b. For pet preference to be completely uninfluenced by gender, about how many girls
would have to prefer each type of pet? Explain.
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Lesson 1
practice
Antonio surveyed 60 of his classmates about their participation in school
activities as well as whether they have a part-time job. The results are shown in
the two-way frequency table below. Use the table to complete the exercises.
1. Complete the table by finding the row totals, column totals, and grand total.
Activity
Clubs
Only
Sports
Only
Both
Neither
Yes
12
13
16
4
No
3
5
5
2
Job
Total
Total
2. Create a two-way relative frequency table using decimals.
Activity
Job
Clubs
Only
Sports
Only
Both
Neither
Total
Yes
No
Total
© Houghton Mifflin Harcourt Publishing Company
3. Give each relative frequency as a percent.
a. The joint relative frequency of students surveyed who participate in school
clubs only and have part-time jobs:
b. The marginal relative frequency of students surveyed who do not have a
part-time job:
c. The conditional relative frequency that a student surveyed participates
in both school clubs and sports, given that the student has a
part-time job:
4. Discuss possible influences of having a part-time job on participation in school
activities. Support your response with an analysis of the data.
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Lesson 1
10-1
Name ________________________________________
Name Class Date __________________
Date Class __________________
Practice
Chapter
Additional
Organizing Practice
and Displaying Data
2
The owner of an ice cream shop conducted a survey regarding
customers’ favorite flavors. The owner asked 80 randomly
selected customers whether they preferred vanilla, chocolate, or
strawberry. The results are shown in the frequency table below.
Flavor
Frequency
Vanilla
22
Chocolate
34
Strawberry
24
Total
80
1. Convert this table to a relative frequency table that uses decimals.
Flavor
Relative
Frequency
Vanilla
Chocolate
Strawberry
Total
2. Convert this table to a relative frequency table that uses percents.
Flavor
Relative
Frequency
Vanilla
Chocolate
Strawberry
Total
© Houghton Mifflin Harcourt Publishing Company
3. The owner also recorded the gender of each customer. The results
are shown in the two-way frequency table below. Complete the table.
Gender/
Flavor
Vanilla
Chocolate
Strawberry
Male
10
20
16
Female
12
14
8
Total
Total
4. Create a two-way relative frequency table for the data in Exercise 3.
Gender/
Flavor
Vanilla
Chocolate
Strawberry
Total
Male
Female
Total
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
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Lesson 1
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class __________________
Problem
Solving
Problem
Solving
Chapter
Organizing and Displaying Data
2
A mobile phone company conducted a survey regarding how people
communicate with their friends. The company asked 200 randomly
selected customers whether they preferred texting, talking, or emailing.
The results are shown in the frequency table below.
Communication
Frequency
Text
116
Talk
54
Email
30
Total
200
1. Convert this table to a relative frequency table that uses decimals.
Communication
Relative
Frequency
Text
Talk
Email
Total
2. The company also recorded the gender of each customer. The results are
shown in the two-way frequency table below. Complete the table.
Gender/
Communication
Text
Talk
Email
Male
75
36
14
Female
41
18
16
Total
Total
Gender/
Communication
Text
Talk
Email
Total
Male
Female
Total
Select the best answer.
4. Find the conditional relative frequency
that a person surveyed prefers talking,
given that the person is male.
5. Find the conditional relative frequency that
a person surveyed is female, given that the
person prefers texting.
A 0.6
C 0.288
F
1
3
B 0.112
D 0.24
G
8
15
H
75
116
J
41
116
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 10
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Sec6:140
Lesson 1
Holt Algebra 1
© Houghton Mifflin Harcourt Publishing Company
3. Create a two-way relative frequency table for the data in Problem 2.