CHAPTER 3 Describing Relationships

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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: 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 Measuring Linear Association: Correlation A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. Linear relationships are important because a straight line is a simple pattern that is quite common. Unfortunately, our eyes are not good judges of how strong a linear relationship is. The correlation r measures the direction and strength of the linear relationship between two quantitative variables. r is always a number between -1 and 1 r > 0 indicates a positive association. r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 towards -1 or 1. The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship. The Practice of Statistics, 5 th Edition 3

4 Back to the track! 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: (a) Make a scatterplot of the relationship between the sprint time and the long jump measure. Use stat/calc on the calculator to find the correlation. (b) (c) Interpret the value of r in context. What effect does the point highlighted in red have on the correlation? Explain. (a) Stat/Calc/LinReg(8)/Xlist/Ylist/calc. The correlation is r = (b) The correlation of r = 0.84 confirms what we see in the scatterplot there is a strong negative association between sprint time and long-jump distance. (c) Because the highlighted point is a little higher than expected based on the pattern of the rest of the data, it makes the correlation closer to 0. The correlation without this point included is r = The Practice of Statistics, 5 th Edition 4

5 Measuring Linear Association: Correlation The Practice of Statistics, 5 th Edition 5

6 Calculating Correlation The formula for r is a bit complex. It helps us to see what correlation is, but in practice, you should use your calculator or software to find r. How to Calculate the Correlation r Suppose that we have data on variables x and y for n individuals. The values for the first individual are x 1 and y 1, the values for the second individual are x 2 and y 2, and so on. The means and standard deviations of the two variables are x-bar and s x for the x-values and y-bar and s y for the y-values. The correlation r between x and y is: r = 1 é æ ê ç n -1ë ê è r = 1 n -1 å x 1 - x s x æ ç è x i - x s x ö æ y - y ö 1 ç ø è s y ø + æ x - x ö æ 2 ç y - y ö 2 è s ç x ø è s y ø æ x - x ö æ n ç y - y ö ù n è s ç x ø è s ú y ø û ú ö æ y - y ö i ç ø è s y ø The Practice of Statistics, 5 th Edition 6

7 Facts About Correlation How correlation behaves is more important than the details of the formula. Here are some important facts about r. 1. Correlation makes no distinction between explanatory and response variables. 2. r does not change when we change the units of measurement of x, y, or both. 3. The correlation r itself has no unit of measurement. Cautions: Correlation requires that both variables be quantitative. Correlation does not describe curved relationships between variables, no matter how strong the relationship is. Correlation is not resistant. r is strongly affected by a few outlying observations. Correlation is not a complete summary of two-variable data. The Practice of Statistics, 5 th Edition 7

8 Correlation Practice Check Your Understanding, p.153 For each graph, estimate the correlation r and interpret it in context. The correlation is about r = 0.9. This indicates that there is a strong, positive linear relationship between the number of boats registered in Florida and the number of manatees killed. The Practice of Statistics, 5 th Edition 8

9 Correlation Practice Check Your Understanding, p.153 For each graph, estimate the correlation r and interpret it in context. The correlation is about r = 0.5. This indicates that there is a moderate, positive linear relationship between the number of named storms predicted and the actual number of named storms. The Practice of Statistics, 5 th Edition 9

10 Correlation Practice Check Your Understanding, p.153 For each graph, estimate the correlation r and interpret it in context. The correlation is about r = 0.3. This indicates that there is a weak, positive linear relationship between the healing rate of the two front limbs of the newts. The Practice of Statistics, 5 th Edition 10

11 Correlation Practice Check Your Understanding, p.153 For each graph, estimate the correlation r and interpret it in context. The correlation is about r = 0.1. This indicates that there is a weak, negative linear relationship between last year s percent return and this year s percent return in the stock market. The Practice of Statistics, 5 th Edition 11

12 Correlation Practice Check Your Understanding, p.153 For each graph, estimate the correlation r and interpret it in context. (a) The correlation is about r = 0.9. This indicates that there is a strong, positive linear relationship between the number of boats registered in Florida and the number of manatees killed. (b) The correlation is about r = 0.5. This indicates that there is a moderate, positive linear relationship between the number of named storms predicted and the actual number of named storms. (c) The correlation is about r = 0.3. This indicates that there is a weak, positive linear relationship between the healing rate of the two front limbs of the newts. (d) The correlation is about r = 0.1. This indicates that there is a weak, negative linear relationship between last year s percent return and this year s percent return in the stock market. The Practice of Statistics, 5 th Edition 12

13 Scoring Essays Peter and Elaine are both English teachers at a high school. Students think that Elaine is a harder grader, so Elaine and Peter decide to grade the same 10 essays and see how their scores compare. The correlation was r = 0.98, but Elaine s scores were about 5 points lower, on average. This can be seen in the scatterplot below, which includes the line y = x. Problem: Is Elaine a harder grader? If both teachers gave an essay the same score, the point for that essay would be on the y = x line. All the points are below the y = x line, however, which indicates that Elaine scored each essay lower than Peter did. The Practice of Statistics, 5 th Edition 13

14 Scatterplots and Correlation Section Summary In this section, we learned how 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 14

15 PAGE , 18, 22, 26 HOMEWORK The Practice of Statistics, 5 th Edition 15

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