1. Chapter 7 Scatterplots, Association, and Correlation

2. Slide 7- 2 Looking at Scatterplots Scatterplots may be the most common and most effective display for data. Look for patterns, trends, relationships, and possible outliers Best way to picture between two variables

3. Slide 7- 3 Looking at Scatterplots (cont.) When looking at scatterplots, we will look for direction, form,, and unusual features Direction: A pattern that runs from the upper left to the lower right is said to have a direction A trend running the other way has a direction

4. Slide 7- 4 Looking at Scatterplots (cont.) This example shows a negative association between central pressure and maximum hurricane wind speed As the central pressure, the maximum wind speed

5. Slide 7- 5 Looking at Scatterplots (cont.) Form: If there is a straight line () relationship, it will appear as a cloud or swarm of points stretched out in a generally consistent, straight form.

6. Slide 7- 6 Looking at Scatterplots (cont.) Form: If the relationship isn’t straight while still increasing or decreasing steadily we can usually find ways to make it more straight

7. Slide 7- 7 Looking at Scatterplots (cont.) Form: If the relationship curves sharply, the methods of this book cannot really help us

8. Slide 7- 8 Looking at Scatterplots (cont.) Strength: At one extreme, the points appear to follow a stream At the other extreme, the points appear as a vague cloud with no discernable trend or

9. Slide 7- 9 Looking at Scatterplots (cont.) Unusual features: Look for the unexpected Look for any outliers standing away from the overall pattern of the scatterplot Clusters or subgroups should also raise questions

10. Slide 7- 10 Roles for Variables Need to determine which of the two quantitative variables goes on the x-axis and which on the y- axis This determination is made based on the roles played by the variables When the roles are clear, the explanatory or variable goes on the x-axis, and the variable goes on the y-axis

11. Slide 7- 11 Data collected from students in Statistics classes included their heights (in inches) and weights (in pounds): There is a positive association Fairly straight form Seems to be a high outlier Correlation

12. Slide 7- 12 How strong is the association between weight and height of Statistics students? Units should not matter when quantifying strength A scatterplot of heights (in centimeters) and weights (in kilograms) doesn’t change the Correlation (cont.)

13. Slide 7- 13 Correlation (cont.) both variables and write the coordinates of a point as (z x, z y ) Removes the units from the data Here is a scatterplot of the standardized weights and heights:

14. Slide 7- 14 Correlation (cont.) The linear pattern seems steeper in the plot than in the scatterplot That’s because we made the scales of the axes the Equal scaling gives a neutral way of drawing the scatterplot and a fairer impression of the

15. Slide 7- 15 Correlation (cont.) A numerical measurement of the strength of the linear relationship between the explanatory and response variables For the students’ heights and weights, the correlation is 0.644 Formula:

16. Slide 7- 16 Correlation Conditions Correlation measures the strength of the linear association between two quantitative variables. Before you use correlation, you must check several conditions: Quantitative Variables Condition Straight Enough Condition Outlier Condition

17. Slide 7- 17 Correlation Conditions (cont.) Quantitative Variables Condition: Correlation applies only to variables Don’t apply correlation to categorical data Need to know the variables’ units and what they measure

18. Slide 7- 18 Correlation Conditions (cont.) Straight Enough Condition: You can calculate a correlation coefficient for any pair of variables Correlation measures the strength only of the linear association Results will be misleading if the relationship is not linear

19. Slide 7- 19 Correlation Conditions (cont.) Outlier Condition: Outliers can distort the correlation It can even change the direction of the correlation coefficient Switch from negative to positive, or When you see an outlier report the correlations with and without that point

20. Slide 7- 20 Correlation Properties The sign of a correlation coefficient gives the direction of the association Correlation is always between Correlation close or equal to -1 or +1 indicates a linear relationship A correlation near zero corresponds to a linear association.

21. Slide 7- 21 Correlation Properties (cont.) Correlation treats x and y The correlation of x with y is the same as the correlation of y with x Correlation has Correlation is not affected by changes in the center or scale of either variable Correlations depend only on

22. Slide 7- 22 Correlation Properties (cont.) Correlation measures the strength of the linear association between the two variables Variables can have a strong association but still have a small correlation if the association isn’t linear Correlation is sensitive to

23. Slide 7- 23 Correlation ≠ Causation Whenever we have a strong correlation, it is tempting to explain it by imagining that the predictor variable has the response Scatterplots and correlation coefficients prove causation Watch out for A hidden variable that stands behind a relationship and determines it by simultaneously affecting the other two variables

24. Slide 7- 24 Correlation Tables Compute the correlations between every pair of variables in a dataset and arrange these correlations in a table

25. Slide 7- 25 Straightening Scatterplots If a scatterplot shows a bent form that consistently increases or decreases, we can often straighten the form of the plot by one or both variables Transforming the data can straighten the scatterplot’s form

26. Slide 7- 26 What Can Go Wrong? Don’t say “correlation” when you mean “association” Don’t confuse “correlation” with “causation” Don’t correlate variables Be sure the association is linear Beware of outliers

27. Slide 7- 27 What Can Go Wrong? (cont.) Don’t assume the relationship is linear just because the correlation coefficient is high R = 0.979, but the relationship is actually bent

28. Slide 7- 28 What have we learned? We examine scatterplots for direction, form, strength, and unusual features Although not every relationship is linear, when the scatterplot is straight enough, the is a useful numerical summary The sign of the correlation tells us the of the association The magnitude of the correlation tells us the of a linear association Shifting, or scaling the data, standardizing, or swapping the variables has no effect on the numerical value

29. Exercise 7.38 Fast food is often considered unhealthy because much of it is high in both fat and calories. But are the two related? Here are the fat contents and calories of several brands of burgers. Analyze the association between fat content and calories. Slide 7- 29 Fat (g)1931343539 43 Calories410580590570640680660

30. Exercise 7.38 (cont.) Let us first plot the data Y-axis? X-axis? Slide 7- 30

31. Slide 7- 31 Exercise 7.38 (cont.)

32. Slide 7- 32

33. Exercise 7.38 (cont.) Slide 7- 33

34. Exercise 7.38 (cont.) Slide 7- 34 From what we learned in Chapter 4:

35. Slide 7- 35 Exercise 7.38 (cont.)

36. Slide 7- 36 Exercise 7.38 (cont.)

37. Slide 7- 37 Exercise 7.38 (cont.)

38. Slide 7- 38