Which of the following is an advantage of the $r$ (correlation coefficient) over the $\mathrm{cov}$ (covariance) when measuring the relationship between two variables?

• A. The $r$ (correlation coefficient) always has a larger magnitude than the $\mathrm{cov}$ (covariance).
• B. The $r$ (correlation coefficient) is unaffected by outliers in the data.
• C. The $r$ (correlation coefficient) can only be used for variables measured in the same units.
• D. The $r$ (correlation coefficient) is unitless, allowing for direct comparison across different datasets.

Answer: (D)

Verified step by step guidance

1. Understand that covariance measures the joint variability of two variables but its value depends on the units of the variables, making it difficult to compare across different datasets or variables with different units.
2. Recognize that the correlation coefficient standardizes the covariance by dividing it by the product of the standard deviations of the two variables, which removes the units and scales the measure between -1 and 1.
3. Recall the formula for the correlation coefficient: \(\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}\), where \(\text{Cov}(X,Y)\) is the covariance, and \(\sigma_X\), \(\sigma_Y\) are the standard deviations of \(X\) and \(Y\) respectively.
4. Note that because the correlation coefficient is unitless, it allows for direct comparison of the strength and direction of relationships across different datasets or variables measured in different units.
5. Conclude that this unitless property is a key advantage of the correlation coefficient over covariance, making it more interpretable and comparable.