1. Introduction A correlation between two events simply means that there is a consistent relationship between two events, and that a change in one event implies that the other event will change according to the linear relationship. A correlation does not imply causation, or that a change in one event causes the change in the second event. Because many factors influence changes in events, it is very difficult to prove causation, sometimes referred to as a causal relationship. For example, researchers gathered the data on the next slide comparing the hours of television watched per day by viewers and each viewer’s weight in pounds. 1 4.3.3: Distinguishing Between Correlation and Causation

2. Introduction, continued 2 4.3.3: Distinguishing Between Correlation and Causation Weight in pounds Hours of television watched per day

3. Introduction, continued The correlation coefficient, r, between the two events is 0.994. According to this data, a person’s weight could be estimated based on the hours of television he or she watches each day. This is due to the strong correlation between the two events. Can the researchers conclude that watching television is responsible for increased weight, or that watching television causes a person to gain weight? 3 4.3.3: Distinguishing Between Correlation and Causation

4. Introduction, continued The correlation does not give any information about why weight increases when a person watches more television. If the data is accurate, it is likely that there is a factor not included in the data set that is associated with both events. For example, a person who watches more television may spend less time exercising and more time sitting. Or, a person who watches more television may eat less healthy foods than someone who spends less time watching television. 4 4.3.3: Distinguishing Between Correlation and Causation

5. Introduction, continued There may be many contributing factors that explain how this relationship works. The important note is that watching television by itself cannot cause weight gain. It is also possible that a correlation appears to exist because of a small sample size or a poorly selected sample. Knowing the difference between correlation and causation is particularly important when reading advertisements or other persuasive materials. Think closely about outside factors that may have influenced both events and led to a strong correlation before believing that one event is responsible for the change in another event. 5 4.3.3: Distinguishing Between Correlation and Causation

6. Key Concepts Correlation does not imply causation. If a change in one event is responsible for a change in another event, the two events have a causal relationship. Outside factors may influence and explain a strong correlation between two events. Use a calculator to find the correlation coefficient. 6 4.3.3: Distinguishing Between Correlation and Causation

7. Key Concepts, continued On a TI-83/84: Step 1: Set up the calculator to find correlations. Press [2nd], then [CATALOG] (the “0” key). Scroll down and select DiagnosticOn, then press [ENTER]. (This step only needs to be completed once; the calculator will stay in this mode until changed in this menu.) Step 2: To calculate the correlation coefficient, first enter the data into a list. Press [2nd], then L1 (the “1” key). Scroll to enter data sets. Press [2nd], then L2 (the “2” key). Enter the second event in L2. 7 4.3.3: Distinguishing Between Correlation and Causation

8. Key Concepts, continued Step 3: Calculate the correlation coefficient. Press [STAT], then select CALC at the top of the screen. Scroll down to 8:LinReg(a+bx), and press [ENTER]. The r value (the correlation coefficient) is displayed along with the equation. 8 4.3.3: Distinguishing Between Correlation and Causation

9. Key Concepts, continued On a TI-Nspire: Step 1: Go to the lists and spreadsheet page. The icon looks like a table. Step 2: Enter the data into the first column underneath the shaded row, pressing [enter] after each data value. Step 3: Use the nav pad to arrow up to the first row below the shaded row and then arrow over to the right so that you are in the second column. Enter the data values, pressing [enter] after each data value. 9 4.3.3: Distinguishing Between Correlation and Causation

10. Key Concepts, continued Step 4: Press the [menu] key. Step 5: Arrow down to 4: Statistics and press the center click key. Step 6: Press the center click key again to select 1: Stat Calculations. Step 7: Choose 3: Linear Regression (mx+b). Step 8: At the X List field, press [clear] and then type in “a[]”. To type “[]”, press the [ctrl] key and then the [(] key. Step 9: Press [tab] to go the Y List field and type in “b[]”. 10 4.3.3: Distinguishing Between Correlation and Causation

11. Key Concepts, continued Step 10: Press [tab] to go the Results field and check that results are listed in “c[]”. If not, change them. Step 11: Press [tab] to “OK” and press the center click key. Step 12: Arrow down until you see the “r” and look to the right. The number to the right is the correlation coefficient, r. 11 4.3.3: Distinguishing Between Correlation and Causation

12. Common Errors/Misconceptions assuming that a strong correlation indicates that one event causes another failing to consider outside factors that may influence the strength of the correlation between two events 12 4.3.3: Distinguishing Between Correlation and Causation

13. Guided Practice Example 1 Alex coaches basketball, and wants to know if there is a relationship between height and free throw shooting percentage. Free throw shooting percentage is the number of free throw shots completed divided by the number of free throw shots attempted: 13 4.3.3: Distinguishing Between Correlation and Causation

14. Guided Practice: Example 1, continued Alex takes some notes on the players in his team, and records his results in the tables on the next two slides. What is the correlation between height and free throw shooting percentage? Alex looks at his data and decides that increased height causes a reduced free throw shooting percentage. Is he correct? 14 4.3.3: Distinguishing Between Correlation and Causation

15. Guided Practice: Example 1, continued 15 4.3.3: Distinguishing Between Correlation and Causation Height in inchesFree throw %Height in inchesFree throw % 75287625 75227042 67307247 8067924 71436923 67407627 76107228 (continued)

16. Guided Practice: Example 1, continued 16 4.3.3: Distinguishing Between Correlation and Causation Height in inchesFree throw %Height in inchesFree throw % 76337513 76257130 67546829 795 14 67557825 7825

17. Guided Practice: Example 1, continued 1.Create a scatter plot of the data. Let the x-axis represent height in inches and the y-axis represent free throw shooting percentage. 17 4.3.3: Distinguishing Between Correlation and Causation

18. Guided Practice: Example 1, continued 18 4.3.3: Distinguishing Between Correlation and Causation Free throw percentage Height in inches 0

19. Guided Practice: Example 1, continued 2.Analyze the scatter plot, and describe any relationship between the two events. As height increases, free throw shooting percentage decreases. It appears that there is a weak negative linear correlation between the two events. 19 4.3.3: Distinguishing Between Correlation and Causation

20. Guided Practice: Example 1, continued 3.Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section. r = –0.727 20 4.3.3: Distinguishing Between Correlation and Causation

21. Guided Practice: Example 1, continued 4.Describe the correlation between the two events. –0.727 is close to –1. There is a negative linear correlation between the events. 21 4.3.3: Distinguishing Between Correlation and Causation

22. Guided Practice: Example 1, continued 5.Consider the causal relationship between the two events. Determine if it is likely that height is responsible for the decrease in free throw shooting percentage. Even if there is a correlation between height and free throw percentage, it is not likely that height causes a basketball player to have more difficulty making free throw shots. 22 4.3.3: Distinguishing Between Correlation and Causation

23. Guided Practice: Example 1, continued If two equally skilled players were of different heights, would you expect one of them to make fewer free throws based only on his or her height? What about a very tall player who spends more time practicing free throws than a very short player? What if the sample size is too small to gather data that’s true for the larger population? There is most likely not a causal relationship between height and free throw percentage. 23 4.3.3: Distinguishing Between Correlation and Causation ✔

24. Guided Practice: Example 1, continued 24

25. Guided Practice Example 2 Mr. Gray’s students are interested in learning how studying can improve test performance. Mr. Gray provides students with practice problems related to particular tests. The class records the number of practice problems completed and the score on that related test in the tables on the next two slides. What is the correlation between the number of practice problems completed and the test score? Is there a causal relationship between the number of practice problems completed and the test score? 25 4.3.3: Distinguishing Between Correlation and Causation

26. Guided Practice: Example 2, continued 26 4.3.3: Distinguishing Between Correlation and Causation Problems completed Test score, out of 100 points Problems completed Test score, out of 100 points 10563050 40706058 100839074 405411089 0453059 505813095 90721046 1509710075 (continued)

27. Guided Practice: Example 2, continued 27 4.3.3: Distinguishing Between Correlation and Causation Problems completed Test score, out of 100 points Problems completed Test score, out of 100 points 11072055 1007418096 12090150100 130993067 1601003056 0492050 6059

28. Guided Practice: Example 2, continued 1.Create a scatter plot of the data. Let the x-axis represent the number of completed practice problems and the y-axis represent the test score. 28 4.3.3: Distinguishing Between Correlation and Causation

29. Guided Practice: Example 2, continued 29 4.3.3: Distinguishing Between Correlation and Causation Test scoreCompleted problems

30. Guided Practice: Example 2, continued 2.Analyze the scatter plot, and describe any relationship between the two events. As the number of completed practice problems increases, the test score also increases. The shape of the graph is approximately a line, and it appears there is a positive linear correlation between the number of practice problems completed and the test score. 30 4.3.3: Distinguishing Between Correlation and Causation

31. Guided Practice: Example 2, continued 3.Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section. r = 0.942 31 4.3.3: Distinguishing Between Correlation and Causation

32. Guided Practice: Example 2, continued 4.Describe the correlation between the two events. 0.942 is close to 1. There is a strong positive linear correlation between the number of practice problems completed and the test score. 32 4.3.3: Distinguishing Between Correlation and Causation

33. Guided Practice: Example 2, continued 5.Consider the causal relationship between the two events. Determine if it is likely that the number of practice problems completed is responsible for increased test scores. Note any other factors that could influence test scores. 33 4.3.3: Distinguishing Between Correlation and Causation

34. Guided Practice: Example 2, continued There is a strong correlation between the two events. A score on a test is related to a student’s knowledge of the test content, and a student’s ability to use content to solve problems. Completing practice problems allows students to develop skills directly related to test performance, and although there are other factors that are related to test performance, it is likely that there is a causal relationship between the number of practice problems completed and the test score. 34 4.3.3: Distinguishing Between Correlation and Causation ✔

35. Guided Practice: Example 2, continued 35