Interpreting Patterns in Scatter Plots Learning Goals In this lesson, you will: Interpret patterns in a scatter plot. Determine if a pattern in a scatter plot has a linear relationship. Identify potential outliers in a scatter plot. Key Terms independent variable (explanatory variable) dependent variable (response variable) association linear association cluster positive association negative association outlier For many East Coast residents visiting sunny Southern California for the first time, a trip to a white sandy beach and warm water is usually on the agenda. However, the ocean temperature in the Pacific Ocean along California s coast is part of a cold current. This means that the average ocean temperature rarely breaks 70ºF! The world s oceans and seas fall into one of two categories: cold currents and warm currents. Generally, the warmest bodies of water are found near the equator while many of the cold currents are closer to the earth s poles. Why does it seem that warm currents are near the equator while cold currents seem to be closer to the North and South Poles? 2011 Carnegie Learning 15.2 Interpreting Patterns in Scatter Plots 807
Problem 1 A table is given that describes the temperature of ocean water at different depths. Students identify the independent variable (depth) and the dependent variable (temperature). They will create a scatter plot and conclude that as the depth of the water increases, the temperature decreases. The data is clustered and displays a negative association. Additional scatter plots are given comparing: the height and weight of soccer players, wait time and movie attendance, student grades for math and art, temperatures in Fahrenheit and Celsius, and the height and time of a ball tossed into the air. Finally, students will analyze each scatter plot. Problem 1 Ocean Temperatures As you learned previously, a two-data variable set is a data set in which two separate characteristics are measured for a person or a thing. Sometimes, the two-data variable set is designated as two different variables: the independent variable and the dependent variable. The independent variable is the variable whose value is not determined by another variable. Generally, the independent variable is represented by the x-coordinate. The dependent variable is the variable whose value is determined by an independent variable. Generally, this dependent variable is represented by the y-coordinate. 1. Erica, who is an oceanographer, is measuring the temperature of the ocean at different depths. Her results are listed in the table. Depth (m) Temperature (ºF) 100 200 300 400 500 600 700 800 900 76 73 70 66 61 56 52 48 43 a. Identify the independent and dependent variables in Erica s data table. The independent variable is depth in meters, and the dependent variable is temperature in degrees Fahrenheit. Grouping Ask a student to read the information and definitions before Question 1 aloud. Then discuss the information as a class. Have students complete Questions 1 through 3 with a partner. Then share the results as a class. Share Phase, Question 1 Does the depth of the ocean depend on the temperature of the ocean, or does the temperature of the ocean depend on the depth of the ocean? b. Create a scatter plot using the data Erica gathered for the ocean temperatures at different depths. Temperature ( o F) 80 76 72 68 64 60 56 52 48 44 40 0 Ocean Temperature at Different Depths 0 100 200 300 400 500 600 700 800 900 1000 Depth of Water (meters) What does the highest point mean with respect to the problem situation? What does the lowest point mean with respect to the problem situation? 808 Chapter 15 Data Displays and Analysis
Share Phase, Questions 2 and 3 As the depth of the water increases, what happens to the temperature of the water? As the temperature of the water decreases, what happens to the temperature of the water? Grouping Ask a student to read the information and definitions after Question 3 aloud. Then complete Questions 4 and 5 as a class. 2. Explain the meaning of the point (400, 66). At a depth of 400 meters, the water temperature is 66 F. 3. What relationship does the scatter plot show between the depth of the ocean water and the temperature of the water? As the depth of the water increases, the temperature decreases. As you have experienced, scatter plots can be great tools to identify patterns in a two-variable data set. Sometimes, these patterns or relationships are called associations. One common pattern that exists in data is when the points on a scatter plot form a linear association. A linear association occurs when the points on the scatter plot seem to form a line. In most cases the points will not form a perfect line, but they are clustered. When data values are clustered, the data values are arranged in such a way that as you look at the graph from left to right, you can imagine a line going through the scatter plot with most of the points being clustered close to the line. 4. Explain how there seems to be a linear association between the depth of the ocean water and the water temperature. The points are clustered in a way that seems to form a line. If the two variables have a linear association, you can then determine the type of association between two variables. You can typically look for a pattern in the dependent variable on the y-axis as the independent variable on the x-axis increases from left to right. The two variables have a positive association if, as the independent variable increases, the dependent variable also increases. If the dependent variable decreases as the independent variable increases, then the two variables have a negative association. Once you identify the pattern for two variables with a linear relationship, you can state the association between the two variables. 2011 Carnegie Learning 15.2 Interpreting Patterns in Scatter Plots 809
5. Describe the type of association that exists between the depth of the ocean water and the water temperature. State the association in terms of the variables. There is a negative association between the depth of the ocean water, and the water temperature. As the depth of the water increases, the temperature decreases. Grouping Have students complete Question 6 with a partner. Then share the results as a class. 6. Analyze each scatter plot shown. Then, identify the following for each: Identify the independent and dependent variables. Determine whether the scatter plots show a linear association. If there is a linear relationship, determine whether it has a positive association or a negative association. State the association in terms of the variables. a. Height and Weight of Soccer Players Weight Height The height of the soccer players is the independent variable, and the weight is the dependent variable. There is a linear association between the two variables. There appears to be a positive association between the two variables. As the height increases the weight also increases. 810 Chapter 15 Data Displays and Analysis
Share Phase, Question 6 Which variable appears on the x-axis? Is the independent variable or the dependent variable usually associated with the x-axis? Which variable appears on the y-axis? Is the independent variable or the dependent variable usually associated with the y-axis? Does the data appear to cluster from the upper left to the lower right or the lower left to the upper right? Data clustered from the upper left to the lower right has what type of association? Data clustered from the lower left to the upper right has what type of association? b. Wait Time and Movie Attendance Movie Attendance Time Waiting in Line Time waiting in line is the independent variable and movie attendance is the dependent variable. There is a linear association between the two variables. There is a negative association between the two variables. As the time waiting in line increases the movie attendance decreases. c. Student Grades in Homeroom 6 100 Grade in Art 90 80 70 60 50 50 60 70 80 90 100 Grade in Math 2011 Carnegie Learning Grade in math is the independent variable and grade in art is the dependent variable. There does not appear to be a linear association. Because there is not a linear association, there cannot be a positive or negative association. 15.2 Interpreting Patterns in Scatter Plots 811
d. Temperature in Fahrenheit and Celsius Degrees in Celsius Degrees in Fahrenheit Temperature, in degrees Fahrenheit, is the independent variable, and temperature in degrees Celsius is the dependent variable. There is a linear association between the two variables. There is a positive association between the two variables. As the degrees in Fahrenheit increase, the degrees in Celsius increase. e. Height of Ball Tossed into Air Height Time Time is the explanatory and height is the response variable. There is not a linear association. Because there is not a linear association, there cannot be a positive or negative association. 812 Chapter 15 Data Displays and Analysis
2011 Carnegie Learning Problem 2 A scatter plot that displays the fat and calories in 11 different foods is given. Students use the scatter plot to determine the independent and dependent variables. They then describe the linear association of the data, complete a table of values, and identify a possible outlier. Grouping Ask a student to read the information and definition aloud. Then complete Question 1 as a class. Discuss Phase, Question 1 What does the highest point mean with respect to the problem situation? What does the lowest point mean with respect to the problem situation? As the number of fat grams increases, what happens to the number of calories? As the number of fat grams decreases, what happens to the number of calories? Does the data appear to cluster from the upper left to the lower right or the lower left to the upper right? Problem 2 Other Patterns in Scatter Plots Another pattern that can occur in a scatter plot is an outlier. An outlier for two-variable data is a point that varies greatly from the overall pattern of the data. 1. The scatter plot shows the fat and calories in 11 different foods. Calories 650 550 450 350 250 0 Fat and Calories in Various Foods 0 10 15 20 25 Fat (grams) a. Determine the independent and dependent variables in the two-variable data set. The independent variable is fat in grams of each food, and the dependent variable is the calories of each food. b. Does there appear to be a linear association between the fat and calories of the foods? Answers will vary. 15.2 Interpreting Patterns in Scatter Plots 813
Grouping Have students complete Questions 2 through 6 with a partner. Then share the responses as a class. 2. Complete a table of values for the points displayed in the scatter plot, beginning with the item with the lowest amount of fat. Fat (g) Calories Share Phase, Questions 2 through 6 Which column in the table is in numerical order? What does it represent with respect to the problem situation? Can a data set have more than one outlier? Explain. Does this data set appear to have more than one outlier? Explain. As the number of outliers in a data set increase, what happens to the type of association? 10 300 11 275 12 325 13 350 15 400 15 450 17 500 20 600 22 575 23 625 25 300 3. Do any of the points appear to vary greatly from the other points? If so, place a star around any outliers in the scatter plot and identify the outlier in the table. 4. Explain why the point (25, 300) is a potential outlier. The point (25, 300) is a potential outlier because the amount of calories in that particular food is much less than other foods with a greater number of fat grams. 5. Examine the values in the table. How can you determine that (25, 300) is a possible outlier? Most of the values in the calorie column increase as I proceed down the column. However, 300 is much less than 600, even though it occurs after 600. 814 Chapter 15 Data Displays and Analysis
6. Place your finger on top of the point (25, 300) and examine the scatter plot. What do you notice? By eliminating the outlier, there is much more of a linear relationship. 2011 Carnegie Learning Talk the Talk Students explain the difference between a positive association and a negative association in a two-variable set. They will explain how to identify an outlier in a two-variable set. Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class. Share Phase, Question 1 Explain what a graph might look like if there is not a positive or negative association of a two-variable data set. Describe an example of a two-variable data set that does not have a positive or negative association. Talk the Talk 1. Explain the difference between a positive association and a negative association of a two-variable data set. A positive association occurs when both the independent variable and the dependent variable increase as the independent variable moves to the right on the scatter plot. A negative association occurs when the independent variable increases in value, but the dependent variable decreases in value as the independent variable moves to the right on the scatter plot. 2. Explain how you can identify an outlier in a two-variable data set. Do the data need to have a linear association? I can identify an outlier in a two-variable set if there is linear association. If there is a data value that does not cluster around or near a line where the other data values are plotted, this data point can be considered an outlier. Be prepared to share your solutions and methods. 15.2 Interpreting Patterns in Scatter Plots 815
Follow Up Assignment Use the Assignment for Lesson 15.2 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson 15.2 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter 15. Check for Students Understanding The height and weight of 12 people are described in the table of values. Height (inches) Weight (pounds) 68 140 66 130 65 122 67 142 72 162 63 165 60 104 69 145 64 121 67 135 70 156 61 108 816 Chapter 15 Data Displays and Analysis
1. Does the table appear to have an outlier? Explain. The point (63, 165) is a possible outlier because the weight seems too large for the height. 2. Use a graphing calculator to graph the scatter plot. Sketch the scatter plot on the coordinate plane shown. Then, verify your answer to Question 1. Yes. The point (63, 165) is an outlier. Weight (pounds) y 165 150 135 120 105 90 75 60 45 30 15 0 10 20 30 40 50 60 70 Height (inches) x 2011 Carnegie Learning 15.2 Interpreting Patterns in Scatter Plots 816A
816B Chapter 15 Data Displays and Analysis