CHAPTER 3 Describing Relationships

Transcription

1 CHAPTER 3 Describing Relationships 3.1 Scatterplots and Correlation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers

2 Scatterplots and Correlation Learning Objectives After this section, you should be able to: IDENTIFY explanatory and response variables in situations where one variable helps to explain or influences the other. MAKE a scatterplot to display the relationship between two quantitative variables. DESCRIBE the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot. INTERPRET the correlation. UNDERSTAND the basic properties of correlation, including how the correlation is influenced by outliers USE technology to calculate correlation. EXPLAIN why association does not imply causation. The Practice of Statistics, 5 th Edition 2

3 Explanatory and Response Variables Most statistical studies examine data on more than one variable. In many of these settings, the two variables play different roles. A response variable measures an outcome of a study. An explanatory variable may help explain or influence changes in a response variable. Note: In many studies, the goal is to show that changes in one or more explanatory variables actually cause changes in a response variable. However, other explanatory-response relationships don t involve direct causation. The Practice of Statistics, 5 th Edition 3

4 Weight and Height Tim wants to know if there is a relationship between height and weight. Kelly wants to know if she can predict a student s weight from his or her height. Information about height is easier to obtain than information about weight! Problem: For each student, identify the explanatory and response variables, if possible. Solution: Tim is just interested in the relationship between the two variables, so there is no clear explanatory or response variable. Kelly is treating a student s height as the explanatory variable and the student s weight as the response variable. The Practice of Statistics, 5 th Edition 4

5 Displaying Relationships: Scatterplots A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph. How to Make a Scatterplot 1. Decide which variable should go on each axis. Remember, the explanatory variable goes on the X-axis! 2. Label and scale your axes. 3. Plot individual data values. The Practice of Statistics, 5 th Edition 5

6 Track and field day! Each member of a small statistics class ran a 40-yard sprint and then did a long jump (with a running start). The table below shows the sprint time (in seconds) and the long-jump distance (in inches) Sprint time (s) Long-jump distance (in) Problem: Make a scatterplot of the relationship between the sprint time and the long jump measure. The Practice of Statistics, 5 th Edition 6

7 Describing Scatterplots To describe a scatterplot, follow the basic strategy of data analysis from Chapters 1 and 2. Look for patterns and important departures from those patterns. How to Examine a Scatterplot As in any graph of data, look for the overall pattern and for striking departures from that pattern. You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship. An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship. The Practice of Statistics, 5 th Edition 7

8 Describing Scatterplots Two variables have a positive association when above-average values of one tend to accompany above-average values of the other and when below-average values also tend to occur together. Two variables have a negative association when above-average values of one tend to accompany below-average values of the other. Strength Describe the scatterplot. Direction Form There is a moderately strong, negative, curved relationship between the percent of students in a state who take the SAT and the mean SAT math score. Further, there are two distinct clusters of states and two possible outliers that fall outside the overall pattern. The Practice of Statistics, 5 th Edition 8

9 Example: Describing a scatterplot Direction: In general, it appears that teams that score more points per game have more wins and teams that score fewer points per game have fewer wins. We say that there is a positive association between points per game and wins. Form: There seems to be a linear pattern in the graph (that is, the overall pattern follows a straight line). Strength: Because the points do not vary much from the linear pattern, the relationship is fairly strong. There do not appear to be any values that depart from the linear pattern, so there are no outliers. The Practice of Statistics, 5 th Edition 9

10 Track and field day! Problem: Describe what the scatterplot in the previous example reveals about the relationship between sprint time and long-jump distances for students in the statistics class. Solution Direction: Students who took longer to run 40 yards tended to have shorter long-jump distances, so there is a negative association between sprint times and long-jump distances. Form: There seems to be a linear pattern in the graph. Strength: Because the points don t vary too much from the linear pattern, the association is strong. Outliers: There don t seem to be any outliers in this scatterplot. The Practice of Statistics, 5 th Edition 10

11 Salads at McDonald s McDonald s restaurants offer a variety of salads. The table below lists 10 different salads, along with the amount of sodium (in mg) and the amount of fat (in grams). source: Problem: Make a scatterplot to display the relationship between the amount of sodium and the amount of fat in salads from McDonald s. Describe what you see. Salad Sodium Fat Southwest Salad with Grilled Chicken Southwest Salad with Crispy Chicken Southwest Salad without chicken Bacon Ranch Salad with Grilled Chicken Bacon Ranch Salad with Crispy Chicken Bacon Ranch Salad without chicken Caesar Salad with Grilled Chicken Caesar Salad with Crispy Chicken Caesar Salad without chicken Side Salad 10 0 The Practice of Statistics, 5 th Edition 11

12 Salads at McDonald s (Continued) There is a positive association between sodium and fat salads with more sodium tend to have more fat. The overall association is nonlinear, as the pattern does not follow a straight line. However, the association is fairly strong as the points do not deviate much from the nonlinear form. Finally, there are three distinct clusters of points, formed by salads with no chicken (lower-left), salads with grilled chicken (lower-right), and salads with crispy chicken (upper-right). Within each cluster there is a positive, linear association between sodium and fat. Make a scatterplot with calculator: Key sodium into L1 and fat into L2. 2 nd Stat plot, choose first graph type, x-list L1 and y-list L2. Graph, Zoom 9 Trace to see points The Practice of Statistics, 5 th Edition 12

13 PAGE 159 1, 4, 6, 8, 14 Homework The Practice of Statistics, 5 th Edition 13