Lesson 3-I ~ Bivariate Data And Frequency Tables 37

BIVARIATE DATA AND FREQUENCY TABLES

LESSON 3-I

When finding the heights of basketball players or listing scores on a test, you are working with single-variable data, also called univariate data. The purpose of univariate data is to describe one set of data. Two-variable data, also called bivariate data, looks at the relationship between two different sets of data. The purpose of bivariate data is to explain relationships or causes. For example, is there a relationship between the amount of absences a student has and the grade a student earns in class? Is there a relationship between people who own dogs and the number of walks they take?

The amount of absences a student has and the grade a student earns in class are examples of numerical data. To see this relationship, create a scatter plot with x-values representing the number of absences and y-values representing the grade earned in class as a percent. Then, determine if there is a correlation between the data.

Absencesx

Gradey

0 950 860 911 851 901 1001 882 873 863 903 707 759 72

10 6520 50

Whether or not a person owns a dog or takes a walk are examples of categorical data. Categorical data is data collected in the form of words and frequencies. Frequency is the number of times an item occurs in a data set.

An example of a category is “dog owner.” The number of times someone answers he or she is a dog owner would be the frequency of dog owners in the sample. A frequency table shows the number of times each category is represented. When looking at bivariate categorical data, you can use a two-way frequency table to see if there is a relationship between two types of categorical data. Scatter plots are not helpful because there are no ordered pairs to plot.

Absences

Gra

de

38 Lesson 3-I ~ Bivariate Data And Frequency Tables

A survey was conducted asking 80 people if they own a dog and if they have taken a walk for exercise within the past two days. The results are below.

◆ 35 people owned a dog and 45 people did not own dogs.◆ Of the 35 people who owned dogs, 15 had walked for exercise in the last two days.◆ Of the 45 people who did not own dogs, 25 had walked for exercise in the last two days.

This information can be organized in a two-way frequency table. Determine the number of dog owners who had walked and not walked as well as the number of non-dog owners who had walked and not walked.

Dog

Ow

ners

Yes NoYes 15 20No 25 20

Walk

Marsha wanted to investigate a relationship between getting a flu shot and getting the flu. She asked fifty people if they had gotten a flu shot and whether or not they had been sick with the flu.

Her data showed that 40 people had gotten a flu shot and 10 had not.Of the 40 with a flu shot, 2 had been sick with the flu.Of the 10 without a flu shot, 7 had been sick with the flu.

Construct a two-way frequency table showing this information.

Determine the categories. "Flu Shot" and "Sick with Flu"

Construct a two-way frequency Yes NoYesNoFl

u Sh

ot

Sick with Flu

table.

Determine the frequencies in Yes No

Yes 2 38No 7 3Fl

u Sh

ot

Sick with Flueach pair of categories.

EXAMPLE 1

solution

Lesson 3-I ~ Bivariate Data And Frequency Tables 39

Consider the two-way frequency table showing dog owners and whether or not the dog owner walked for exercise in the last two days.

Yes NoYes 15 20No 25 20

Dog

Ow

ner Walk

To better understand how the data may be related, you can use relative frequencies. A relative frequency is the ratio of the observed frequency to the total number of frequencies in the experiment or survey. In the case of the dog owners and walking for exercise, 80 people were surveyed. To find the relative frequencies, find the ratio of each two-way category to 80.

Yes No

Yes 15 __ 80 = 3 __ 16 ≈ 0.19 20 __ 80 = 1 _ 4 = 0.25

No 25 __ 80 = 5 __ 16 ≈ 0.31 20 __ 80 = 1 _ 4 = 0.25Dog

Ow

ner

Walk

The relative frequencies show: Yes No

Yes

≈ 19%The probability a person owns a dog and has

walked is about 19%

25%The probability a person owns a dog and has not walked is 25%

No

≈ 31%The probability a person does not own a dog and has walked is

about 31%

25%The probability a person does not

own a dog and has not walked is 25%

In a two-way frequency table you can also look at conditional frequencies. A conditional frequency looks at the ratio within one category. For example, to look at whether a dog owner is more likely to have walked for exercise in the past week, look at the ratio of each frequency for dog owner and compare it to the total dog owners surveyed.

Walk

Dog

Ow

ner

40 Lesson 3-I ~ Bivariate Data And Frequency Tables

Yes NoYes 15 20No 25 20

Dog

Ow

ner Walk

◆ The conditional frequency for a dog owner who has walked is 15 __ 35 = 3 _ 7 ≈ 0.43

◆ The conditional frequency for a dog owner who has not walked is 20 __ 35 = 4 _ 7 ≈ 0.57

This means there is a 43% chance a dog owner has walked in the past two days and a 57% chance a dog owner has not walked in the past two days. There is a slightly higher chance if a person is a dog owner that they have not walked in the past two days.

a. Find the relative frequencies for the two-way table in Example 1. b. Explain one observation from the relative frequencies.c. What is the conditional frequency that someone with a flu shot will not get sick?

a. Yes No

Yes 2 38No 7 3Fl

u Sh

ot

Sick with Flu

Determine the number of people in the survey 2 + 38 + 7 + 3 = 50

Find the ratio of each frequency to the total number surveyed.

Yes No

Yes 2 __ 50 = 1 __ 25 = 0.04 38 __ 50 = 19 __ 25 = 0.76

No 7 __ 50 = 0.14 3 __ 50 = 0.06Flu

Shot

Sick with Flu

b. One relative frequency shows 76% of those surveyed had a flu shot and did not get sick. It is likely a person asked will have had a flu shot and not gotten sick. c. Find the number of people who had a flu shot. 2 + 38 = 40

Find the ratio of the frequency to 40. 38 __ 40 = 19 __ 20 = 0.95

According to this data, there is a 95% chance you will not get the flu if you have a flu shot. The relationship suggests one should get a flu shot.

EXAMPLE 2

solutions

Lesson 3-I ~ Bivariate Data And Frequency Tables 41

EXERCISES

1. Alex wrote the following information about the two-way frequency table. Which statement is incorrect? Find the statement and correct it.

◆ 60 people who chose hot dogs used a bun. ◆ 18 people who chose hot dogs did not use a bun. ◆ 2 people who did not choose hot dogs used a bun. ◆ 40 people who did not choose hot dogs did not use a bun. ◆ There were 100 people surveyed at a picnic about whether they chose to eat a hot dog and whether they chose to eat a bun.

Read each two-way frequency table. a. Find the total number of people surveyed. b. Write each pair of categories with their frequency. 2.

Yes NoYes 50 15No 5 20C

hore

s

Curfew 3. Yes No

Yes 70 15No 5 30Ta

ller t

han

6 fe

et

Basketball Player

4. Two hundred random seniors from a high school were asked if they had passed Algebra II and if they had passed a college readiness exam. ◆ Of the 120 who had passed Algebra II, 110 had passed the exam. ◆ Of the 80 who had not passed Algebra II, 30 had passed the exam. Construct a two-way frequency table showing this information.

5. Ninety people were randomly selected to answer whether or not they liked to fly and whether or not they had a passport. ◆ Of the 70 people who liked to fly, 60 of them had a passport. ◆ Fifteen people who did not like to fly had a passport. Construct a two-way frequency table showing this information.

6. One hundred twenty people were randomly selected to answer whether they preferred country music or rap music and whether they preferred the color green or purple. ◆ Of the 60 people who preferred country music, 35 preferred the color purple. ◆ Thirty of the people who preferred rap music preferred the color purple. Construct a two-way frequency table showing this information.

7. Eighty baseball games for one team were randomly selected from a season of games. Whether the team scored four or more runs and whether the team won was recorded. ◆ Of the 60 wins, in 45 games the team had scored four or more runs. ◆ The team had scored four or more runs in 8 of the losses. Construct a two-way frequency table showing this information.

Yes NoYes 60 18No 2 40H

ot D

og

Bun

42 Lesson 3-I ~ Bivariate Data And Frequency Tables

8. Which of the two-way frequency tables in Exercises 4–7 shows the least relationship between the two sets of data?

9. In a two-way frequency table, what is the difference between a relative frequency and a conditional frequency?

For Exercises 10–11: a. Make a two-way relative frequency table. Round to the nearest hundredth. b. Explain one observation from the relative frequencies.

10. Yes No

Yes 20 50No 5 35C

ell P

hone

Home Phone 11. Yes No

Yes 20 30No 30 20Pl

ay G

olf

Play Tennis

12. Write the conditional frequencies for people with cell phones in Exercise 10. If a person has a cell phone are they likely to also have a home phone? Explain.

13. Write the conditional frequencies for people who play golf in Exercise 11. If a person plays golf do they likely also play tennis? Explain.

For Exercises 14–15, use relative frequencies and/or conditional frequencies to explain whether or not there is a relationship between the bivariate data. Label your work and explain your thinking. 14.

Yes NoYes 15 5No 5 15H

iker

Camper 15. Yes No

Yes 8 22No 15 30Play

Pia

no

Play Guitar