Learn on PengienVision, Algebra 1Chapter 11: Statistics

Lesson 5: Two-Way Frequency Tables

In this Grade 11 Algebra 1 lesson from enVision Chapter 11, students learn how to organize categorical data in two-way frequency tables and interpret joint frequencies, marginal frequencies, joint relative frequencies, marginal relative frequencies, and conditional relative frequencies. Students practice calculating these values to identify trends and make inferences, such as comparing preferences across groups while accounting for differences in sample size. The lesson builds skills in data analysis and statistical reasoning using real-world survey and sports contexts.

Section 1

Two-way frequency tables

Property

Patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
A two-way frequency table is a convenient way of summarizing such data.
The table is 'two-way' because each bivariate datum is composed of an ordered pair of realizations from two categorical random variables.
The table is a 'frequency' table because the cell entries count the number of subjects that fall into each combination of categories.

Examples

  • A survey asks 100 students if they prefer pizza or burgers, and if they prefer soda or water. The results are organized in a 2×22 \times 2 table showing how many students fall into each of the four combinations (e.g., Pizza and Soda).
  • A clinic records the pet type (Dog, Cat, Bird) and reason for visit (Check-up, Sick). A 3×23 \times 2 two-way frequency table is used to count how many dogs came for a check-up, how many cats were sick, etc.
  • 8th graders are surveyed on their favorite school subject (Math, English, Science) and their after-school activity (Sports, Music, None). The data is tallied in a 3×33 \times 3 table to see if there are associations between subject preference and activities.

Explanation

When you have data in categories (like 'male'/'female' or 'cat'/'dog'), you can't make a scatter plot. A two-way table sorts this data into a grid, showing the counts for each combination, which helps you spot patterns.

Section 2

Defining Joint Frequency

Property

A joint frequency is the count of outcomes that satisfy two different categorical criteria simultaneously.
These are the values found in the body of a two-way frequency table, where a specific row and a specific column intersect.

Examples

Consider the table showing student transportation methods:
| | Walk | Bus | Total |
| :--- | :---: | :---: | :---: |
| Boys | 8 | 12 | 20 |
| Girls | 10 | 15 | 25 |
| Total | 18 | 27 | 45 |

  • The joint frequency of boys who walk to school is 88.
  • The joint frequency of girls who take the bus is 1515.

Explanation

A joint frequency shows how many data points fall into a specific combination of two categories. For instance, in a table categorizing students by gender and transportation, a joint frequency tells you the exact number of boys who walk or the number of girls who take the bus. These frequencies are the core data entries "inside" the table, not the totals in the margins. Each joint frequency represents the intersection of one row category and one column category.

Section 3

Marginal frequencies

Property

We can calculate the marginal frequencies (the count of the occurrence of one variable at a time).
Marginal frequency is the total count for a specific category in a two-way table. It is found by summing the joint frequencies in a particular row or column.
It is always the case that the sum the marginal frequencies of a given variable equals the total number of subjects, so adding marginal frequencies provides a useful check for mistakes.

Examples

  • In a survey of 60 students about after-school clubs (Sports/Chess) and grade (7th/8th), the table shows 25 7th graders chose sports. If 40 students chose sports in total, 40 is the marginal frequency for the 'Sports' category.
  • A table tracks 100 pet owners with dogs or cats who live in houses or apartments. The sum of the 'Dog' row is its marginal frequency, representing the total number of dog owners surveyed.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Two-way frequency tables

Property

Patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
A two-way frequency table is a convenient way of summarizing such data.
The table is 'two-way' because each bivariate datum is composed of an ordered pair of realizations from two categorical random variables.
The table is a 'frequency' table because the cell entries count the number of subjects that fall into each combination of categories.

Examples

  • A survey asks 100 students if they prefer pizza or burgers, and if they prefer soda or water. The results are organized in a 2×22 \times 2 table showing how many students fall into each of the four combinations (e.g., Pizza and Soda).
  • A clinic records the pet type (Dog, Cat, Bird) and reason for visit (Check-up, Sick). A 3×23 \times 2 two-way frequency table is used to count how many dogs came for a check-up, how many cats were sick, etc.
  • 8th graders are surveyed on their favorite school subject (Math, English, Science) and their after-school activity (Sports, Music, None). The data is tallied in a 3×33 \times 3 table to see if there are associations between subject preference and activities.

Explanation

When you have data in categories (like 'male'/'female' or 'cat'/'dog'), you can't make a scatter plot. A two-way table sorts this data into a grid, showing the counts for each combination, which helps you spot patterns.

Section 2

Defining Joint Frequency

Property

A joint frequency is the count of outcomes that satisfy two different categorical criteria simultaneously.
These are the values found in the body of a two-way frequency table, where a specific row and a specific column intersect.

Examples

Consider the table showing student transportation methods:
| | Walk | Bus | Total |
| :--- | :---: | :---: | :---: |
| Boys | 8 | 12 | 20 |
| Girls | 10 | 15 | 25 |
| Total | 18 | 27 | 45 |

  • The joint frequency of boys who walk to school is 88.
  • The joint frequency of girls who take the bus is 1515.

Explanation

A joint frequency shows how many data points fall into a specific combination of two categories. For instance, in a table categorizing students by gender and transportation, a joint frequency tells you the exact number of boys who walk or the number of girls who take the bus. These frequencies are the core data entries "inside" the table, not the totals in the margins. Each joint frequency represents the intersection of one row category and one column category.

Section 3

Marginal frequencies

Property

We can calculate the marginal frequencies (the count of the occurrence of one variable at a time).
Marginal frequency is the total count for a specific category in a two-way table. It is found by summing the joint frequencies in a particular row or column.
It is always the case that the sum the marginal frequencies of a given variable equals the total number of subjects, so adding marginal frequencies provides a useful check for mistakes.

Examples

  • In a survey of 60 students about after-school clubs (Sports/Chess) and grade (7th/8th), the table shows 25 7th graders chose sports. If 40 students chose sports in total, 40 is the marginal frequency for the 'Sports' category.
  • A table tracks 100 pet owners with dogs or cats who live in houses or apartments. The sum of the 'Dog' row is its marginal frequency, representing the total number of dog owners surveyed.