Car rideshare services are a popular option for people needing to move about in large cities. The scatterplot shows the distances of trips and fares, in dollars, for an adult living in a city over a period of a month. The value of r for the scatterplot is 0.950.
How would the value of the correlation coefficient change if the fares were plotted on the x-axis and distances on the y-axis?
A. The value of the correlation coefficient would be –0.950.
B. The value of the correlation coefficient would not change.
C. The value of the correlation coefficient would be closer to 0.
D. The value of the correlation coefficient would be closer to 1.
The Correct Answer and Explanation is:
The correct answer is B. The value of the correlation coefficient would not change.
Explanation:
The correlation coefficient (denoted as r) measures the strength and direction of the linear relationship between two variables. In this case, the scatterplot shows the relationship between the distance of trips and the corresponding fares for a rideshare service. The value of r = 0.950 indicates a very strong positive linear relationship between these two variables, meaning that as the distance increases, the fare tends to increase as well.
Key Characteristics of the Correlation Coefficient:
- Symmetry: The value of the correlation coefficient does not depend on which variable is plotted on the x-axis or the y-axis. The correlation is a measure of the strength and direction of the relationship between the two variables, not their positions on the axes. Whether you plot the fares on the x-axis and distances on the y-axis, or vice versa, the correlation coefficient remains the same because it reflects the strength of the relationship, not the specific roles of the variables.
- Formula for r: The correlation coefficient r is calculated using the formula that involves the covariance of the two variables divided by the product of their standard deviations. The formula does not change if you switch the variables on the axes because both the covariance and the standard deviations are unaffected by the order in which the variables are plotted. As a result, the value of r remains constant.
- Interpretation of r: Since the magnitude of r is the same regardless of which variable is plotted on the x-axis or y-axis, a positive correlation of r = 0.950 would still indicate a very strong positive linear relationship whether you plot fares versus distance or distance versus fares.
Conclusion:
In conclusion, the value of the correlation coefficient will not change if you swap the axes. The relationship between the variables remains the same, and therefore, the correlation coefficient stays at 0.950, making B the correct answer.