Topics
5. Probability
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United States of America
Grades 9-12

5.02 Relative frequency tables

Lesson
Practice

Introduction

Relative frequency tables appeared in 8th grade, and we used our knowledge of two-way tables to extend our understanding of types of frequencies in lesson  5.01 Two-way frequency tables  . In this lesson, we will look at more specific row-relative and column-relative conditional frequencies and determine associations in the data.

Relative frequency tables

The joint relative frequency is the ratio of a joint frequency to the total number of data points, and the marginal relative frequency is the ratio of the marginal frequency to the total number of data points. When these relative frequencies are displayed in a two-way table, we call it a two-way relative frequency table.

Exploration

Consider the following two-way tables based on the survey from lesson  5.01 Two-way frequency tables 

A two way table titled Two way frequency table with 3 columns titled High top sneakers, Low top sneakers, and Total, and 3 rows titled Online shopping, In store shopping, and Total. The data is as follows: Online shopping: High top sneakers, 23, Low top sneakers, 12, Total, 35; In store shopping: High top sneakers, 7, Low top sneakers, 33, Total, 40; Total: High top sneakers, 30, Low top sneakers, 45, Total, 75
A two way table titled Two way relative frequency table with 3 columns titled High top sneakers, Low top sneakers, and Total, and 3 rows titled Online shopping, In store shopping, and Total. The data is as follows: Online shopping: High top sneakers, 31%, Low top sneakers, 16%, Total, 47%; In store shopping: High top sneakers, 9%, Low top sneakers, 44%, Total, 53%; Total: High top sneakers, 40%, Low top sneakers, 60%, Total, 100%
A two way table titled Row relative frequency table with 3 columns titled High top sneakers, Low top sneakers, and Total, and 3 rows titled Online shopping, In store shopping, and Total. The data is as follows: Online shopping: High top sneakers, 66%, Low top sneakers, 34%, Total, 100%; In store shopping: High top sneakers, 17.5%, Low top sneakers, 82.5%, Total, 100%; Total: High top sneakers, 40%, Low top sneakers, 60%, Total, 100%
A two way table titled Column relative frequency table with 3 columns titled High top sneakers, Low top sneakers, and Total, and 3 rows titled Online shopping, In store shopping, and Total. The data is as follows: Online shopping: High top sneakers, 77%, Low top sneakers, 27%, Total, 47%; In store shopping: High top sneakers, 23%, Low top sneakers, 73%, Total, 53%; Total: High top sneakers, 100%, Low top sneakers, 100%, Total, 100%
  1. How do you think the two-way frequency table is related to the two-way relative frequency table?
  2. How do you think the two-way frequency table is related to the row-relative frequency table?
  3. How do you think the two-way frequency table is related to the column-relative frequency table?

A two-way relative frequency table shows the proportion or percentage of each entry out of the total number of data points.

A row-relative frequency table shows the proportion or percentage of each entry out of the total number of data points in the row. Each row will total to 100 \%.

A column-relative frequency table shows the proportion or percentage of each entry out of the total number of data points in the column. Each column will total to 100 \%.

Row-relative and column-relative frequencies are conditional relative frequencies.

Conditional relative frequency

The ratio of a joint relative frequency and a marginal relative frequency. Equivalently, the ratio of a relative frequency and a marginal frequency.

For categorical data, we can recognize possible associations and trends in the data by using relative frequencies to make conjectures.

Association

A way to describe the form, direction or strength of the relationship between the two variables in a bivariate data set.

An association exists between two variables if the row or column conditional relative frequencies are different for different rows or columns of the table. The greater the differences, the stronger the association.

There is likely no association between the variables if the conditional relative frequencies are nearly equal for all categories.

Examples

Example 1

This two-way table represents survey results from a sample of college students regarding their preferred place to study:

LibraryStudent UnionCoffee ShopHomeTotal
Sociology Majors43215922145
Math Majors34161541106
Total77377463251
a

Construct the relative frequency table for the data.

Worked Solution
Create a strategy

Relative frequency is found by dividing each entry in the table by the total population, which in this case is 251 students.

Apply the idea
LibraryStudent UnionCoffee ShopHomeTotal
Sociology Majors{\frac{43}{251}=17\%}\frac{21}{251}=8\%\frac{59}{251}=24\%\frac{22}{251}=9\%\frac{145}{251}=58\%
Math Majors\frac{34}{251}=14\%\frac{16}{251}=6\%\frac{15}{251}=6\%{\frac{41}{251}=16\%}\frac{106}{251}=42\%
Total\frac{77}{251}=31\%\frac{37}{251}=15\%\frac{74}{251}=29\%\frac{63}{251}=25\%{\frac{251}{251}=100\%}
Reflect and check

The joint relative frequencies may not appear to equal the marginal relative frequencies due to rounding error.

b

Construct the row-relative frequency table for the data.

Worked Solution
Create a strategy

Row-relative frequency is a conditional frequency found by dividing each entry in the table by its row marginal frequency.

Apply the idea
LibraryStudent UnionCoffee ShopHomeTotal
Sociology Majors{\frac{43}{145}=30\%}\frac{21}{145}=14\%\frac{59}{145}=41\%{\frac{22}{145}=15\%}{\frac{145}{145}=100\%}
Math Majors\frac{34}{106}=32\%\frac{16}{106}=15\%\frac{15}{106}=14\%\frac{41}{106}=39\%\frac{106}{106}=100\%
Total\frac{77}{251}=31\%\frac{37}{251}=15\%\frac{74}{251}=29\%\frac{63}{251}=25\%\frac{251}{251}=100\%
Reflect and check

In row-relative frequency tables the final column will always be 100\%.

c

Construct the column-relative frequency table for the data.

Worked Solution
Create a strategy

Column-relative frequency is a conditional frequency found by dividing each entry in the table by its column marginal frequency.

Apply the idea
LibraryStudent UnionCoffee ShopHomeTotal
Sociology Majors\frac{43}{77}=56\%\frac{21}{37}=57\%\frac{59}{74}=80\%\frac{22}{63}=35\%\frac{145}{251}=58\%
Math Majors\frac{34}{77}=44\%\frac{16}{37}=43\%\frac{15}{74}=20\%\frac{41}{63}=65\%\frac{106}{251}=42\%
Total{\frac{77}{77}=100\%}\frac{37}{37}=100\%\frac{74}{74}=100\%{\frac{63}{63}=100\%}{\frac{251}{251}=100\%}
Reflect and check

In column-relative frequency tables the final column will always be 100\%.

Example 2

This two-way table represents the number of traffic violations in Portland, Oregon during the past three weeks:

SpeedingTexting While DrivingTotal
Teenage Drivers168096
Adult Drivers491564
Total6595160
a

Determine which traffic violation is committed most by teenage drivers and what proportion of the tickets for teenagers were of that type.

Worked Solution
Create a strategy

Teenage drivers represent a row category of data, so we need to divide the joint frequencies in the first row by the marginal frequency of the first row. This is called row-relative frequency.

Apply the idea

Speeding violations represent \frac{16}{96} \approx 17\% of teenage traffic violations and texting while driving represents \frac{80}{96} \approx = 83\% of teenage traffic violations. Texting while driving is the most common violation committed by teenage drivers.

Reflect and check

Carefully consider whether the question is answered by a relative frequency, row-relative frequency, or column-relative frequency before answering.

b

Find the percentage of all speeding violations that are committed by adult drivers.

Worked Solution
Create a strategy

Speeding violations represent a column category of data, so we need to divide the joint frequency of an adult committing a speeding violation by the marginal frequency of speeding. This is called the column-relative frequency.

Apply the idea

Adults commit \frac{49}{65} \approx 75\% of speeding violations.

Example 3

Mayra surveyed some students at a local fair. She asked the participants their age and whether they have a cell phone.

Has a cell phoneDoes not have a cell phoneTotal
Ages 10-12132033
Ages 13-15372057
Ages 16-1851960
Total9951150

Describe the association, if any, between a person's age and whether or not they have a cell phone. Justify your response.

Worked Solution
Create a strategy

Construct a relative frequency table to analyze any association between a person's age and whether or not they have a cell phone.

Apply the idea

For this problem, a row-relative frequency table or a column-relative frequency table may help us decide whether there is an association.

Has a cell phoneDoes not have a cell phoneTotal
Ages 10-12 39\%61\%100\%
Ages 13-1565 \%35 \%100\%
Ages 16-1885\%15\%100\%
Total66\%34\%100\%

Based on the row-relative frequency table, we can see that as students get older, a higher percentage have a cell phone. There is an association between the student's age and whether or not they have a cell phone.

Reflect and check

We can confirm that there is an association between the age and having a cell phone because the percentages are significantly different for having a cell phone and not having a cell phone, which suggests that age does have an impact.

Idea summary
  • A row-relative frequency table shows the proportion or percentage of each entry out of the total number of data points in the row
  • A column-relative frequency table shows the proportion or percentage of each entry out of the total number of data points in the column
  • Associations in data or lack thereof can be described justified by comparing how different the percentages are in relative frequency tables.

Outcomes

S.ID.B.5

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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