1. Correlation: How Strong Is the Linear Relationship? Lecture 46 Sec. 13.7 Mon, Apr 30, 2007

2. The Correlation Coefficient The correlation coefficient r is a number between –1 and +1. It measures the direction and strength of the linear relationship.  If r > 0, then the relationship is positive. If r < 0, then the relationship is negative.  The closer r is to +1 or –1, the stronger the relationship.  The closer r is to 0, the weaker the relationship.

3. Strong Positive Linear Association x y In this display, r is close to +1.

4. Strong Positive Linear Association x y In this display, r is close to +1.

5. Strong Negative Linear Association In this display, r is close to –1. x y

6. Strong Negative Linear Association In this display, r is close to –1. x y

7. Almost No Linear Association In this display, r is close to 0. x y

8. Almost No Linear Association In this display, r is close to 0. x y

9. Interpretation of r -0.8-0.20.8010.2

10. Interpretation of r -0.8-0.20.8010.2 Strong Negative Strong Positive

11. Interpretation of r -0.8-0.20.8010.2 Weak Negative Weak Positive

12. Interpretation of r -0.8-0.20.8010.2 No Significant Correlation

13. Correlation vs. Cause and Effect If the value of r is close to +1 or -1, that indicates that x is a good predictor of y. It does not indicate that x causes y (or that y causes x). The correlation coefficient alone cannot be used to determine cause and effect.

14. Mixing Populations Mixing nonhomogeneous groups can create a misleading correlation coefficient. Suppose we gather data on the number of hours spent watching TV each week and the child’s reading level, for 1 st, 2 nd, and 3 rd grade students.

15. Mixing Populations We may get the following results, suggesting a weak positive correlation. Number of hours of TV Reading level 10141822 1.0 2.0 3.0

16. Mixing Populations We may get the following results, suggesting a weak positive correlation. Number of hours of TV Reading level 10141822 1.0 2.0 3.0 r = 0.26

17. Mixing Populations However, if we separate the points according to grade level, we may see a different picture. 1 st grade 2 nd grade 3 rd grade Number of hours of TV Reading level 10141822 1.0 2.0 3.0

18. Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level 10141822 1.0 2.0 3.0 r 1 = -0.35

19. Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level 10141822 1.0 2.0 3.0 r 2 = -0.73

20. Mixing Populations However, if we separate the points according to grade level, we may see a different picture. Number of hours of TV Reading level 10141822 1.0 2.0 3.0 r 3 = -0.52

21. Calculating the Correlation Coefficient There are many formulas for r. The most basic formula is Another formula is

22. Example Consider again the data xy 23 35 59 612 916

23. Example Compute  x,  y,  x 2,  y 2, and  xy. 155 515 282 2545 xyx2x2 y2y2 xy 23496 3592515 59258145 6123614472 91681256144

24. Example Then compute r.

25. TI-83 – Calculating r To calculate r on the TI-83,  First, be sure that Diagnostic is turned on. Press CATALOG and select DiagnosticsOn.  Then, follow the procedure that produces the regression line.  In the same window, the TI-83 reports r 2 and r. Use the TI-83 to calculate r in the preceding example.

26. Example Find the correlation coefficient for the Calorie/Cholesterol data. Calories (x)350290330290320370280290310230 Cholesterol (y)5020451535502025200

27. How Does r Work? Recall the formula We will consider the numerator.

28. How Does r Work? Consider the Subway data: Cal (x)Chol (y) 35050 29020 33045 29015 32035 37050 28020 29025 31020 2300

29. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  x 3505044 29020–16 3304524 29015–16 3203514 3705064 28020–26 29025–16 310204 2300–76

30. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y 350504422 29020–16–8 330452417 29015–16–13 32035147 370506422 28020–26–8 29025–16–3 310204–8 2300–76–28

31. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) 350504422968 29020–16–8128 330452417408 29015–16–13208 3203514798 3705064221408 28020–26–8208 29025–16–348 310204–8–32 2300–76–282128

32. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) 350504422968 29020–16–8128 330452417408 29015–16–13208 3203514798 3705064221408 28020–26–8208 29025–16–348 310204–8–32 2300–76–282128

33. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) 350504422968 29020–16–8128 330452417408 29015–16–13208 3203514798 3705064221408 28020–26–8208 29025–16–348 310204–8–32 2300–76–282128

34. How Does r Work? Consider the Subway data: Cal (x)Chol (y) x –  xy –  y(x –  )(y –  y) 350504422968 29020–16–8128 330452417408 29015–16–13208 3203514798 3705064221408 28020–26–8208 29025–16–348 310204–8–32 2300–76–282128

35. How Does r Work? 200250300350400 Calories Cholesterol 0 10 20 30 50 40

36. How Does r Work? 200250300350400 Calories Cholesterol 0 10 20 30 50 40

37. How Does r Work? 200250300350400 Calories Cholesterol 0 10 20 30 50 40

38. How Does r Work? 200250300350400 Calories Cholesterol 0 10 20 30 50 40positive negative positive

39. How Does r Work? 200250300350400 Calories Cholesterol 0 10 20 30 50 40positive negative positive