Understanding Correlation in Two-Variable Data Analysis
This content discusses calculating the correlation between two quantitative variables, its properties, and how outliers can influence the correlation. It provides a step-by-step guide on how to calculate the correlation coefficient 'r' using z-scores. Additionally, it presents a practical problem of whether touchdowns predict wins in the NFL's NFC West division, illustrating the application of correlation analysis in real-world scenarios.
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Analyzing Two-Variable Data Lesson 2.4 Calculating the Correlation Statistics and Probability with Applications, 3rdEdition Starnes & Tabor Bedford Freeman Worth Publishers
Calculating the Correlation Learning Targets After this lesson, you should be able to: Calculate the correlation between two quantitative variables. Apply the properties of the correlation. Describe how outliers influence the correlation. Statistics and Probability with Applications, 3rdEdition 2 2
Calculating the Correlation In the previous lesson, we learned that the correlation r measures the strength and direction of the linear relationship between two quantitative variables. How to Calculate the Correlation r 1. Find the mean x-bar and the standard deviation sx of the explanatory variable. Calculate the z-score for the value of the explanatory variable for each individual. Find the mean y-bar and the standard deviation sy of the response variable. Calculate the z-score for the value of the response variable for each individual. For each individual, multiply the z-score for the explanatory variable and the z-score for the response variable. Add the z-score products and divide the sum by n 1. 2. 3. 4. Statistics and Probability with Applications, 3rdEdition 3 3
Do touchdowns predict wins? Do touchdowns predict wins? Calculating the correlation Calculating the correlation PROBLEM: The table shows the number of touchdowns and the number of wins for the four teams in the NFC West division of the NFL during a recent season. Calculate the correlation for these data. Team Arizona Cardinals Seattle Seahawks St. Louis Rams San Francisco 49ers Touchdowns 58 49 31 24 Wins 13 10 7 5 Statistics and Probability with Applications, 3rd Edition 4 4
z z 58 40.5 13 8.75 49 40.5 10 8.75 31 40.5 7 8.75 24 40.5 5 8.75 x y x y = = = = = 15.7 3.5 15.7 3.5 15.7 3.5 15.7 3.5 1.11 1.21 0.54 0.36 = 0.61 = 0.5 1.05 1.07 = = = z z x y x y s s Do touchdowns predict wins? Do touchdowns predict wins? Calculating the correlation Calculating the correlation x y Statistics and Probability with Applications, 3rd Edition 5 5
Calculating the Correlation To calculate the correlation, both variables must be quantitative. If one or both of the variables are categorical, we can consider the association between the two variables, but not the correlation. Properties of the Correlation r 1. Correlation makes no distinction between explanatory and response variables. It makes no difference which variable you call x and which you call y in calculating the correlation. Statistics and Probability with Applications, 3rd Edition 6 6
Calculating the Correlation Properties of the Correlation r 2. Because r uses the standardized values of the observations, r does not change when we change the units of measurement of x, y, or both. Measuring foot length and height in inches rather than centimeters does not change the correlation between foot length and height. Statistics and Probability with Applications, 3rd Edition 7 7
Calculating the Correlation Properties of the Correlation r 3. The correlation r has no units of measurement because we are using standardized values in the calculation and standardized values have no units. Statistics and Probability with Applications, 3rd Edition 8 8
Weight training, third rep? Weight training, third rep? Properties of correlation Properties of correlation PROBLEM: The scatterplot below shows the relationship between the amount of weight students from Gary Lang s weight training class can squat and bench press from the example in Lesson 2.2. The correlation is r = 0.939. 450 400 350 Squat weight (pounds) 300 250 200 150 100 100 120 140 160 180 200 Bench press weight (pounds) (a) What would happen to the correlation if squat weight was plotted on the horizontal axis and bench press weight was plotted on the vertical axis? Explain. The correlation would still be r = 0.939, because the correlation makes no distinction between explanatory and response variables. Statistics and Probability with Applications, 3rd Edition 9 9
Weight training, third rep? Weight training, third rep? Properties of correlation Properties of correlation (b) What would happen to the correlation if squat weight was measured in kilograms instead of pounds? Explain. The correlation would still be r = 0.939, because the correlation doesn t change when we change the units of either variable. (c) Egnarts claims that the correlation between squat weight and bench press weight is r = 0.939 pounds. Is this correct? No. The correlation doesn t have units, so including pounds is incorrect. Statistics and Probability with Applications, 3rd Edition 10 10
Calculating the Correlation The formula for correlation involves the mean and standard deviation of both variables. Because the mean and standard deviation aren t resistant to outliers, the correlation isn t resistant to outliers either. Consider a scatterplot in which the majority of points form a positive, linear association. An outlier that is in the same pattern as the rest of the points will make the correlation closer to 1. An outlier that is not in the pattern of the rest of the points will make the correlation closer to 0 or possibly negative. Statistics and Probability with Applications, 3rd Edition 11 11
Smaller person, smaller I.Q.? Smaller person, smaller I.Q.? Outliers and Correlation Outliers and Correlation PROBLEM: The scatterplot below shows the relationship between the height (in inches) and cumulative weighted GPA for a sample of high school students. How do the two points in red (in the upper-right corner of the graph) affect the correlation? Explain. The majority of points shows a moderate negative association. However, because the two points in the upper-right are separated far from the rest of the points in the x direction, they are very influential and will make the correlation closer to 0 and possibly positive. Without these two points, the correlation is clearly negative. Statistics and Probability with Applications, 3rd Edition 12 12
LESSON APP 2.4 Flying dinosaur or early bird? Archaeopteryx is an extinct beast having feathers like a bird, but teeth and a long bony tail like a reptile. Because the known specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. However, if the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the lengths of a pair of bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths (in centimeters) of the femur (a leg bone) and the humerus (a bone in the upper arm) for five specimens that preserve both bones. Statistics and Probability with Applications, 3rd Edition 13 13
LESSON APP 2.4 Flying dinosaur or early bird? 1. Make a scatterplot using length of femur as the explanatory variable. Do you think that all five specimens come from the same species? Explain. 1. Find the correlation r step by step, using the formula on page 121. Explain how your value for r matches your graph in part (1). 1. Suppose that a new fossil was discovered. If the femur is 70 centimeters and the humerus is 40 centimeters, do you think this specimen came from the same species? Explain. 1. What effect will the new fossil have on the correlation? Explain. Statistics and Probability with Applications, 3rd Edition 14 14
LESSON APP 2.4 Flying dinosaur or early bird? Statistics and Probability with Applications, 3rd Edition 15 15
LESSON APP 2.4 Flying dinosaur or early bird? Statistics and Probability with Applications, 3rd Edition 16 16
LESSON APP 2.4 Flying dinosaur or early bird? Statistics and Probability with Applications, 3rd Edition 17 17
Calculating the Correlation Learning Targets After this lesson, you should be able to: Calculate the correlation between two quantitative variables. Apply the properties of the correlation. Describe how outliers influence the correlation. Statistics and Probability with Applications, 3rd Edition 18 18
