Lesson 28 Introduction M.8.25 Scatter lots · Independent ractice Practice 246 ©Curriculum...

Introduction

©Curriculum Associates, LLC Copying is not permitted.240

Use What You Know

Lesson 28 Scatter Plots

Lesson 28Scatter Plots

In previous lessons, you learned how to represent relationships between two variables on the coordinate plane. Take a look at this problem.

Mr. Finley wanted to see if there was a relationship between students’ scores on the final exam in algebra and the amount of time they spent studying for it. Students studied for the test by doing practice problems on a website that recorded the number of minutes they spent studying. He recorded the number of minutes of studying and test scores in a table. Does the table show a relationship?

Use the math you already know to solve the problem.

a. What is the highest score on the exam and the corresponding

studying time?

b. What is the lowest score on the exam and the corresponding

studying time?

c. What is the approximate mean test score and mean studying time,

to the nearest tenth?

d. Are there any ordered pairs that are surprising to you? Explain.

e. Do you think that spending more time studying helped these students do better on the test? Why or why not?

Minutes Studied

Test Score

60 85

65 85

70 90

90 100

88 94

99 98

86 38

99 99

96 95

45 70

30 60

56 72

30 35

40 52

25 30

M.8.25

©Curriculum Associates, LLC Copying is not permitted. 241

Find Out More


A scatter plot is a graph of ordered pairs in the coordinate plane that represents data. Each ordered pair reports information about two variables. Mr. Finley’s data compares minutes a student spent studying (the x-coordinate) with that student’s test score (the y-coordinate). The study of two related variables is called bivariate data analysis.

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Scatter plots show characteristics of data sets that are not easily seen in tables. For example, notice the clustering above. Points outside the main cluster of data stand out. The ordered pair (86, 38) is outside the main cluster. This point might be considered an outlier.

Scatter plots also help reveal the type of association there may be between two variables. The test scores and time spent studying show a positive association because as time spent studying increased, test scores tended to increase as well. A positive association is represented by an upward trend from left to right in the scatter plot. A negative association shows a downward trend from left to right; one variable decreases as the other increases.

In this scatter plot, the trend of the data could be modeled by a straight line through the cluster of points. So, you could describe the relationship shown in the graph as being roughly linear. This is not true for all scatter plots. Some may show nonlinear relationships.

Reflect1 Is it surprising that there is a positive association between the variables in Mr. Finley’s

data set rather than a negative association? Why or why not?

Modeled and Guided Instruction

Learn About


Lesson 28


Identifying Positive and Negative Associations

Read the problem below. Then explore relationships in bivariate data sets.

Mr. Finley wondered if there was a relationship between the number of texts students sent the day before the final exam and their test scores. He asked students to volunteer the data from their cell phones, and then he compared it against their test scores. How can he identify trends?

Model It You can plot the data in a table and scatter plot to show clusters and associations.

Add a horizontal line to represent the mean score and a vertical line to represent the mean number of texts. Look for trends by finding the quadrants that contain large clusters of points.

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Number of Texts

Test

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mean number of texts

mean score on test

Model It You can also cut the scatter plot into vertical slices and compute the mean test score for each slice.

The triangles point to the mean test score for each slice to its right. This representation gives more information about the trend of the data.

Number of Texts Score on Test

50 85

52 85

57 90

38 100

50 94

40 98

75 38

39 99

45 95

67 70

72 60

62 72

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87 30

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©Curriculum Associates, LLC Copying is not permitted. 243Lesson 28 Scatter Plots

Connect It Now you will analyze the data set from the previous page.

2 What is the greatest number of text messages sent? What is the least number sent?

3 Are there any points that lie outside the general trend of the data? Explain. You may want to compare the data on the previous page with the scatter plot in the introduction.

4 Does the scatter plot in the first Model It show a positive association or a negative

association? Explain.

5 How does showing a horizontal line for one mean and a vertical line for the other mean

help support your answer to Problem 4?

6 Describe the trend of the means in the sliced scatter plot, which is in the second Model It. How does the trend of the means help support your answer to Problem 4?

Try It Use what you just learned about associations between variables to solve this problem. Show your work on a separate sheet of either lined paper or graph paper.

7 Describe a situation where a negative association would exist between two variables. Explain why you would expect a negative association. Then make up a data set with 15 ordered pairs and graph them in a scatter plot.

Guided Practice

Practice


Lesson 28



Read the situation described below. Then solve problems 8–10.

Example

Mr. Finley’s class tried a new website to study for the next test. Mr. Finley again compared time spent studying against test scores using the data displays below.

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Sco

re 80

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20

0

40 50 60

Minutes Studied

70 80 90 100

Test

Sco

re

Minutes Studied

100

50

0

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48−59

60−71

72−83

84−95

96−108

Does there seem to be an association between time spent studying on the new website and test scores?

Solution

8 Do you think Mr. Finley should encourage students to study algebra on the website mentioned in the introduction or on this new website?

Solution

Pair/ShareName another real-world pair of variables that might have no association with each other.

Possible answer: There is no clear upward or downward trend in the data displays. Points appear to be randomly scattered throughout the scatter plot. About the same number of points fall within each quadrant. The mean for each slice does not increase or decrease with a consistent trend.

Scatter plots help us decide what kind of association might exist between two variables.

You can cut a scatter plot into four parts by making a horizontal line to show the mean y-value and a vertical line for the mean x-value.


9 Look at the quadrants in the graph on the previous page. What percentage of the points fall in each of the four quadrants? What do these percentages tell you about the association between the two variables?

Show your work.

Solution

10 Which pair of variables would you expect to have a negative association?

A The amount of money you make in a week and the number of hours you work during the week.

B A person’s height and the number of pets he or she owns.

C The speed at which a person drives to work and the time it takes her to get to work.

D The number of friends you have and the number of birthday parties you attend in a year.

Brian chose B as the correct answer. Why is Choice B incorrect?

Pair/ShareWhat would the graph on the previous page look like if two of the quadrants contained 0% of the total number of points?

Pair/ShareHow would you help Brian understand his error?

Find the number of points in each quadrant and the total number of points on the graph. Then divide the number in each quadrant by the total number of points on the graph.

You can often predict if a pair of real-world variables will have positive or negative association by thinking about the context.

Independent Practice

Practice


Lesson 28



Solve the problems.

1 Students at Central Middle School decided to gather survey data from their classmates on a number of different questions. Which pair of variables from the survey would be most likely to have no association with one another?

A number of hours spent studying per week and grade point average

B height in inches and age in months

C number of books owned and number of visits to the doctor’s office in one year

D height and weight

2 The scatter plot below represents 2011 data comparing the average teacher salary and average student/teacher ratios for a number of school districts in the state of Vermont (source: education.vt.gov).

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age

Stud

ent/

Teac

her R

atio

7.5

7.8

9.5

10.5

11.5

12.5

13.5

48 53 58

Average Teacher Salary(thousands of dollars)

63

Choose True or False for each statement.

a. An outlier exists at the point representing a salary of approximately $62,000. True False

b. There is a negative linear association between average salaries and average student/teacher ratios. True False

c. The smallest school district has an average student/ teacher ratio of approximately 8.6. True False

d. There is a positive linear association between average salaries and average student/teacher ratios. True False

e. The school district with an average salary of $48,000 and a student/teacher ratio of 13 is an outlier compared to districts paying less than $48,000. True False

Self Check


Go back and see what you can check off on the Self Check on page 239.

3 Each scatter plot below can be described by one of the following types of association.

Positive linear Negative linear Nonlinear None

Write the correct association below each scatter plot.

4 Invent your own set of bivariate data. Use the blank table and the scatter plot to both create and show your data. The data set should meet the following conditions:

• All x-values must be between 0 and 20; all y-values must be between 0 and 15.

• The association between the variables should be both negative and linear.

• One point should be an outlier; label that point “outlier.”

Explain how your data set meets the above conditions by referring to specific features of the scatter plot and data set.

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Lesson 28 Introduction M.8.25 Scatter lots · Independent ractice Practice 246 ©Curriculum...

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