What is true about r and r^2?

• A. r^2 is the variance of the data
• B. The coefficient of determination r^2 is the square of the correlation coefficient r
• C. r^2 is the covariance
• D. r is the slope of the regression line

Answer: (B)

The key idea is that the correlation r measures the strength and direction of a linear relationship between two variables, while r^2 (the coefficient of determination) tells you what proportion of the variance in one variable is explained by that linear relationship with the other. This makes r^2 the square of r, and it always falls between 0 and 1 (even if r is negative). It’s not the data’s variance (that’s variance), nor the covariance (which involves multiplying deviations without squaring and scaling by standard deviations), and the regression slope is not r itself but r scaled by the ratio of the standard deviations of the two variables. So r^2 being the square of the correlation coefficient is the correct statement. For example, if r = 0.8, r^2 = 0.64, meaning about 64% of the variance in the outcome is explained by the linear relationship with the predictor.