Videos and lessons with examples and solutions to help Grade 8 students learn how to
organize bivariate categorical data into a two-way table. Students calculate row and
column relative frequencies and interpret them in context.

New York State Common Core Math Grade 8, Module 6, Lesson 13.

#### Exercises 1–8

On an upcoming field day at school, the principal wants to provide ice cream during lunch. She will offer three flavors:
chocolate, strawberry, and vanilla. She selected your class to complete a survey to help her determine how much of each
flavor to buy.

1. Answer the following question. Wait for your teacher to count how many students answered for each flavor. Then, record the class totals for each flavor in the chart below.

“Which of the following three ice cream flavors is your favorite: Chocolate, strawberry, or vanilla?”

2. Which ice cream flavor do most students prefer?

3. Which ice cream flavor is preferred by the fewest students?

4. What percentage of students preferred each flavor? Round to the nearest tenth of a percent.

5. Do the numbers in the chart above summarize data on a categorical variable or a numerical variable?

6. Do the students in your class represent a random sample of all students in your school? Why or why not? Discuss this with your neighbor.

7. Is your class representative of all the other classes at your school? Why or why not? Discuss this with your neighbor.

8. Do you think the principal will get an accurate estimate of the proportion of students that prefer each ice cream flavor for the whole school using only your class? Why or why not? Discuss this with your neighbor.

#### Example 1

Students in a different class were asked the same question about their favorite ice cream flavor. The table below shows
the ice cream flavors and the number of students who chose each flavor for that particular class. This table is called a
one-way frequency table because it shows the counts of a univariate categorical variable.

We compute the relative frequency for each ice cream flavor by dividing the count by the total number of observations. Since out of students answered “chocolate,” the relative frequency would be 11/25 = 0.44. This relative frequency shows that 44% of the class prefers chocolate ice cream. In other words, the relative frequency is the proportional value that each category is of the whole.

#### Exercises 9–10

Use the table for the preferred ice cream flavors from the class in Example 1 to answer the following questions.

9. What is the relative frequency for the category “strawberry?”

10. Write a sentence interpreting the relative frequency value in the context of strawberry ice cream preference.

#### Example 2

The principal also wondered if boys and girls have different favorite ice cream flavors. She decided to redo the survey by
taking a random sample of students from the school and recording both their favorite ice cream flavor and their gender.

She asked the following two questions:

• “Which of the following ice cream flavors is your favorite: Chocolate, strawberry, or vanilla?”

• “What is your gender: Male or female?”

The results of the survey are as follows: • Of the 30 students who preferred chocolate ice cream, 22 are males.

• Of the 25 students who preferred strawberry ice cream, 15 are females.

• Of the 27 students who preferred vanilla ice cream, 13 are males.

The values of two variables, which were ice cream flavor and gender, were recorded in this survey. Since both of the variables are categorical, the data are bivariate categorical data.

#### Exercises 11–17

11. Can we display these data in a one-way frequency table? Why or why not?

12. Summarize the results of the second survey of favorite ice cream flavors in the following table:

13. Calculate the relative frequencies for the table above and write them in the table.

Use the relative frequency values in the table to answer the following questions:

14. What is the proportion of the students that prefer chocolate ice cream?

15. What is the proportion of students that are female and prefer vanilla ice cream?

16. Write a sentence explaining the meaning of the approximate relative frequency 0.55.

17. Write a sentence explaining the meaning of the approximate relative frequency 0.10.

#### Example 3

In the previous exercises, you used the total number of students to calculate relative frequencies. These relative
frequencies were the proportion of the whole group who answered the survey a certain way. Sometimes row or column
totals are used to calculate relative frequencies. We call these row relative frequencies or column relative frequencies.
Below is the two-way frequency table for your reference. To calculate “the proportion of male students that prefer
chocolate ice cream,” divide the 22 male students who preferred chocolate ice cream by the total of 45 male students.
This proportion is 22/45 = 0.49. Notice that you used the row total to make this calculation. This is a row relative
frequency.

#### Exercises 18–22

In Exercise 13, you used the total number of students to calculate relative frequencies. These relative frequencies were
the proportion of the whole group who answered the survey a certain way.

18. Suppose you are interested in the proportion of male students that prefer chocolate ice cream. How is this value different from “the proportion of students that are male and prefer chocolate ice cream?” Discuss this with your neighbor.

19. Use the table provided in Example 3 to calculate the following relative frequencies.

a. What proportion of students that prefer vanilla ice cream is female?

b. What proportion of male students prefers strawberry ice cream? Write a sentence explaining the meaning of this proportion in context of this problem.

c. What proportion of female students prefers strawberry ice cream?

d. What proportion of students who prefer strawberry ice cream is female?

20. A student is selected at random from this school. What would you predict this student’s favorite ice cream to be?

Explain why you choose this flavor.

21. Suppose the randomly selected student is male. What would you predict his favorite flavor of ice cream to be?

Explain why you choose this flavor.

22. Suppose the randomly selected student is female. What would you predict her favorite flavor of ice cream to be?

Explain why you choose this flavor.

Lesson 13 Exit Ticket

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Lesson Summary

- Univariate categorical data are displayed in a one-way frequency table.
- Bivariate categorical data are displayed in a two-way frequency table.
- Relative frequency is the frequency divided by a total (frequency/total).
- A cell relative frequency is a cell frequency divided by the total number of observations.
- A row relative frequency is a cell frequency divided by the row total.
- A column relative frequency is a cell frequency divided by the column total.

Lesson 13 Classwork

1. Answer the following question. Wait for your teacher to count how many students answered for each flavor. Then, record the class totals for each flavor in the chart below.

“Which of the following three ice cream flavors is your favorite: Chocolate, strawberry, or vanilla?”

2. Which ice cream flavor do most students prefer?

3. Which ice cream flavor is preferred by the fewest students?

4. What percentage of students preferred each flavor? Round to the nearest tenth of a percent.

5. Do the numbers in the chart above summarize data on a categorical variable or a numerical variable?

6. Do the students in your class represent a random sample of all students in your school? Why or why not? Discuss this with your neighbor.

7. Is your class representative of all the other classes at your school? Why or why not? Discuss this with your neighbor.

8. Do you think the principal will get an accurate estimate of the proportion of students that prefer each ice cream flavor for the whole school using only your class? Why or why not? Discuss this with your neighbor.

We compute the relative frequency for each ice cream flavor by dividing the count by the total number of observations. Since out of students answered “chocolate,” the relative frequency would be 11/25 = 0.44. This relative frequency shows that 44% of the class prefers chocolate ice cream. In other words, the relative frequency is the proportional value that each category is of the whole.

9. What is the relative frequency for the category “strawberry?”

10. Write a sentence interpreting the relative frequency value in the context of strawberry ice cream preference.

She asked the following two questions:

• “Which of the following ice cream flavors is your favorite: Chocolate, strawberry, or vanilla?”

• “What is your gender: Male or female?”

The results of the survey are as follows: • Of the 30 students who preferred chocolate ice cream, 22 are males.

• Of the 25 students who preferred strawberry ice cream, 15 are females.

• Of the 27 students who preferred vanilla ice cream, 13 are males.

The values of two variables, which were ice cream flavor and gender, were recorded in this survey. Since both of the variables are categorical, the data are bivariate categorical data.

12. Summarize the results of the second survey of favorite ice cream flavors in the following table:

13. Calculate the relative frequencies for the table above and write them in the table.

Use the relative frequency values in the table to answer the following questions:

14. What is the proportion of the students that prefer chocolate ice cream?

15. What is the proportion of students that are female and prefer vanilla ice cream?

16. Write a sentence explaining the meaning of the approximate relative frequency 0.55.

17. Write a sentence explaining the meaning of the approximate relative frequency 0.10.

18. Suppose you are interested in the proportion of male students that prefer chocolate ice cream. How is this value different from “the proportion of students that are male and prefer chocolate ice cream?” Discuss this with your neighbor.

19. Use the table provided in Example 3 to calculate the following relative frequencies.

a. What proportion of students that prefer vanilla ice cream is female?

b. What proportion of male students prefers strawberry ice cream? Write a sentence explaining the meaning of this proportion in context of this problem.

c. What proportion of female students prefers strawberry ice cream?

d. What proportion of students who prefer strawberry ice cream is female?

20. A student is selected at random from this school. What would you predict this student’s favorite ice cream to be?

Explain why you choose this flavor.

21. Suppose the randomly selected student is male. What would you predict his favorite flavor of ice cream to be?

Explain why you choose this flavor.

22. Suppose the randomly selected student is female. What would you predict her favorite flavor of ice cream to be?

Explain why you choose this flavor.

Lesson 13 Exit Ticket

1. Explain what the term bivariate categorical data means.

2. Explain how to calculate relative frequency. What is another word for relative frequency?

3. A random group of students is polled about how they get to school. The results are summarized in the table below.

a. Calculate the relative frequencies for the table above. Write them as a percent in each cell of the table. Round to the nearest tenth of a percent.

b. What is the relative frequency for the “Carpool” category? Write a sentence interpreting this value in the context of school transportation.

c. What is the proportion of students that are female and walk to school? Write a sentence interpreting this value in the context of school transportation.

d. A student is selected at random from this school. What would you predict this student’s mode of school transportation to be? Explain.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.