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212 Lesson 5.7 ~ Bivariate Data and Frequency Tables
bivariaTE daTa and frEquEncy TablEs
Lesson 5.7
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
student Grades and Absences
Lesson 5.7 ~ Bivariate Data and Frequency Tables 213
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 noyes 2 38no 7 3Fl
u sh
ot
sick with Flu
each pair of categories.
ExamplE 1
solution
214 Lesson 5.7 ~ Bivariate Data and Frequency Tables
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 determine if there is any relationship between dog owners and whether or not a person walked for exercise in the last two days, 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%.
These percents are close to one another so there is not a clear relationship between the data. 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
Lesson 5.7 ~ Bivariate Data and Frequency Tables 215
yes noyes 15 20no 25 20
dog
ow
ner walk
◆ The conditional frequency for a dog owner who has walked is 15 3 0.4335 7
= ≈
◆ The conditional frequency for a dog owner who has not walked is 20 4 0.5735 7
= ≈
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 showing the possible relationship between those who had a flu shot and those who were sick with the flu. 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.
ExamplE 2
solutions
216 Lesson 5.7 ~ Bivariate Data and Frequency Tables
ExErcisEs
1. Alex wrote the following information about the two-way frequency table. Which statement is incorrect? Find the statement and correct it.
◆ 70 people who chose hot dogs used a bun. ◆ 15 people who chose hot dogs did not use a bun. ◆ 5 people who did not choose hot dogs used a bun. ◆ 30 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.
For exercises 2-3: 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 the team had scored four or more runs in 45 games. ◆ The team had scored four or more runs in 8 of the losses. Construct a two-way frequency table showing this information.
yes noyes 70 15no 5 30h
ot d
og
Bun
Lesson 5.7 ~ Bivariate Data and Frequency Tables 217
8. Which of the two-way frequency tables in exercises 4–7 shows the weakest relationship between the two sets of data? Explain how you know your answer is correct.
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 noyes 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 your reasoning.
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 your reasoning.
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 30Pl
ay P
iano
Play Guitar
rEviEw
describe the possible bias in each survey situation. explain how to modify each situation to eliminate the bias.
16. To collect data on Americans' exercise habits, you survey people going in and out of the local fitness center.
17. To collect data on 8th grade students’ heights, you measure the height of all the boys in your first period class.
218 Lesson 5.7 ~ Bivariate Data and Frequency Tables
18. Pete researched the cost of flying a group of people from Orlando to Boston for a baseball game. He found that as more people were included in the group, the cost per person went down. The data he collected is in the following table.
number of People in Group 1 3 5 6 8 9 10 11 12 13 15
Cost per Person (dollars) 350 325 300 295 265 260 255 250 240 230 215
a. Make a scatter plot of the data. Let x represent the number of people in the group and y represent the cost per person. b. Find the quartiles of the x- and y-values. Draw in horizontal and vertical lines on your graph at those values. c. Write the ordered pairs for the two Q-points which the data follows. d. Find the equation of the line of best fit using the Q-points. Write the values as decimals to the nearest hundredth. e. Use your equation to estimate the cost per person if a group had 25 members. Show all work to support your answer.
tic-tAc-toe ~ cor r e l Ations i n th e re A l wor l d
Find three examples of data sets with correlations in a magazine or newspaper. The three data sets should include at least one example of a positive correlation and one example of a negative correlation. Cut out or make a copy of each data set. Create a scatter plot, draw a line of best fit and describe the correlation of each data set. Explain why you think each data set has the given correlation.
tic-tAc-toe ~ con duct A Bi VA r i Ate su rV e y
Create a survey with two “Yes or No” questions that you could test to see if there is a relationship. For example, if you want to see if there is a relationship between people who have dogs and the number of walks they take, you might ask the following questions:
Do you own a dog? and Have you taken a walk in the past two days?
Survey at least 20 students, asking each student the same two questions, and record the data. Use a two-way frequency table, relative frequencies, and conditional frequencies to analyze your data. Write a 1-page summary of the results from your survey. What conclusions can you draw from your data? Was there any bias in the questions? Did you survey a random sample of students? If you were to do your survey again, would you change anything about the data-collection process?
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