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

IntroductionA 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.

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4.3.3: Distinguishing Between Correlation and Causation

Introduction, continued

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Wei

gh

t in

po

un

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Hours of television watched per day

Introduction, continuedThe 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?

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Introduction, continuedThe 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.

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Introduction, continuedThere 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.

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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.

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Key Concepts, continuedOn 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.

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Key Concepts, continuedStep 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.

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Key Concepts, continuedOn 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.

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Key Concepts, continuedStep 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


Key Concepts, continuedStep 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.

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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

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Guided Practice

Example 1Alex 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:

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Guided Practice: Example 1, continuedAlex 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?

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Guided Practice: Example 1, continued

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Height in inches Free throw % Height in inches Free throw %

75 28 76 25

75 22 70 42

67 30 72 47

80 6 79 24

71 43 69 23

67 40 76 27

76 10 72 28

(continued)


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Height in inches Free throw % Height in inches Free throw %

76 33 75 13

76 25 71 30

67 54 68 29

79 5 79 14

67 55 78 25

78 25


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.

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Fre

e th

row

per

cen

tag

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Height in inches

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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.

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3. Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section.r = –0.727

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4. Describe the correlation between the two events.–0.727 is close to –1. There is a negative linear correlation between the events.

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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.

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Guided Practice: Example 1, continuedIf 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.

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Guided Practice

Example 2Mr. 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?

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Problems completed

Test score, out of 100 points

Problems completed


10 56 30 50

40 70 60 58

100 83 90 74

40 54 110 89

0 45 30 59

50 58 130 95

90 72 10 46

150 97 100 75

(continued)


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Problems completed


Problems completed


110 72 0 55

100 74 180 96

120 90 150 100

130 99 30 67

160 100 30 56

0 49 20 50

60 59


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.

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Test

sco

re

Completed problems


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.

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3. Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section.r = 0.942

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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.

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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.

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Guided Practice: Example 2, continuedThere 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.

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Documents

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