How is Pearson r related to coefficient of determination r^2?

• A. r equals r^2
• B. r^2 equals r^2
• C. r equals square root of r^2
• D. r^2 equals -r

Answer: (C)

Pearson r is a measure of both the strength and direction of a linear relationship between two variables, while the coefficient of determination r^2 tells us the proportion of variance in the outcome that is explained by the predictor in a simple linear regression. They are directly related because r^2 is simply r squared. This means the magnitude of r equals the square root of r^2, and the sign of r indicates the direction of the relationship. In other words, r = ± sqrt(r^2): the positive or negative sign matches how the variables move together, while r^2 itself is always nonnegative and reflects only the strength of the association as a proportion of explained variance. For example, r = 0.6 gives r^2 = 0.36, meaning 36% of the variance is explained; if r = -0.6, r^2 is still 0.36, but r is negative, reflecting an inverse relationship.