Activity
Thinking back to the last activity, you learned about the -value. Write these characteristics about the -value.
The -value is called a correlation coefficient. A correlation coefficient is one way to measure the strength of a linear relationship.
- The sign of is the same as the sign of the slope of the best fit line.
- The values for go from –1 to 1 and include –1 and 1.
- The closer is to 1 or –1, the stronger the linear relationship between the variables.
- The closer is to 0, the weaker the linear relationship between the variables.
Take turns with your partner to match a scatter plot with a correlation coefficient. For each match you find, explain to your partner how you know it’s a match. For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
A
B
C
D
E
F
G
H
Compare your answer:
Scatter Plot F
Compare your answer:
Scatter Plot C
Compare your answer:
Scatter Plot D
Compare your answer:
Scatter Plot G
Compare your answer:
Scatter Plot H
Compare your answer:
Scatter Plot E
Compare your answer:
Scatter Plot B
Compare your answer:
Scatter Plot A
Video: Matching Correlation Coefficients
Watch the following video to learn more about the connections of scatter plots to the correlation coefficient.
Are you ready for more?
Extending Your Thinking
Making a Real-World Connection
Jada wants to know if the speed that people walk is correlated with their texting speed. To investigate this, she measured the distance, in feet, that 5 of her friends walked in 30 seconds and the number of characters they texted during that time. Each of the 5 friends took 4 walks for a total of 20 walks. Here are the results of the first 20 walks.
Distance (feet) | Number of characters texted | Distance (feet) | Number of characters texted |
---|---|---|---|
105 | 14 | 95 | 138 |
125 | 110 | 125 | 110 |
115 | 120 | 160 | 80 |
140 | 98 | 175 | 64 |
145 | 102 | 130 | 106 |
160 | 89 | 140 | 95 |
170 | 72 | 150 | 95 |
140 | 100 | 155 | 90 |
130 | 107 | 160 | 74 |
105 | 113 | 135 | 108 |
Over the next few days, the same 5 friends practiced walking and texting to see if they could walk faster and text more characters. They did not record any more data while practicing. After practicing, each of the 5 friends took another 4 walks. Here are the results of the final 20 walks.
Distance (feet) | Number of characters texted | Distance (feet) | Number of characters texted |
---|---|---|---|
140 | 410 | 165 | 151 |
150 | 155 | 170 | 136 |
160 | 151 | 190 | 143 |
155 | 170 | 205 | 132 |
180 | 125 | 205 | 128 |
205 | 130 | 210 | 140 |
225 | 95 | 215 | 109 |
175 | 161 | 220 | 105 |
195 | 108 | 230 | 126 |
155 | 142 | 225 | 138 |
1. What do you notice about the 2 scatter plots?
Compare your answer:Your answer may vary, but here is a sample.
I noticed that in the second scatter plot, the students traveled longer distances. In addition, the students seemed to type more characters for any given distance in the second scatter plot than in the first scatter plot.
2. Jada noticed that her friends walked farther and texted faster during the last 20 walks than they did during the first 20 walks. Since both were faster, she predicts that the correlation coefficient of the line of best fit for the last 20 walks will be closer to –1 than the correlation coefficient of the line of best fit for the first 20 walks. Do you agree with Jada? Be prepared to show your reasoning.
Compare your answer: You answer may vary, but here is a sample.
I do not agree with Jada. The correlation coefficient deals with the strength and direction of a linear relationship. The comparison bewteen walking distrance and the number of characters texted relates to the slope of the line, because it compares rates of number of characters texted over the distanced walked.
3. Use technology to find an equation of the line of best fit for each data set.
Compare your answer: You answer may vary, but here is a sample.
The line of best fit for scatter plot 1 is . The line of best fit for scatter plot 2 is .
4. The correlation coefficient for the first line of best fit is and the second line of best fit is . Was your answer to question 2 correct? Why do you think the correlation coefficients for the two data sets are so different? Be prepared to show your reasoning.
Compare your answer: You answer may vary, but here is a sample.
I think the correlation coefficients are different because the data in the second scatter plot are more scattered than the data in the first scatter plot. On average, the data in the first scatter plot are much closer to the line of best fit than the data in the second scatter plot.
Self Check
Additional Resources
Correlation Coefficient
Try matching each scatter plot to one of the following correlation coefficients:
, , ,
Scatter plot | Correlation coefficient |
Scatter plot A |
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Scatter plot B |
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Scatter plot C |
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Scatter plot D |
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Try it
Try It: Correlation Coefficient
The scatter plot below displays data on the number of defects per 100 cars and a measure of customer satisfaction (on a scale from 1 to 1,000, with higher scores indicating greater satisfaction) for the 33 brands of cars sold in the United States in 2009.
Which of the following is the value of the correlation coefficient for this data set: , , , or ? Explain why you selected this value.
There is not a strong pattern of a linear relationship with this scatter plot. Even though there is a weak relationship, there is a general pattern that as the number of defects increases, the satisfaction rating tends to decrease. As a result, the value of is negative and close to zero. Based on the values provided, the best estimate is .