Many Faces of the Correlation Coefficient

Abstract

Some selected interpretations of Pearson's correlation coefficient are considered. Correlation may be interpreted as a measure of closeness to identity of the standardized variables. This interpretation has a psychological appeal in showing that perfect covariation means identity up to positive linearity. It is well known that r is the geometric mean of the two slopes of the regression lines. In the 2 x 2 case, each slope reduces to the difference between two conditional probabilities so that r equals the geometric mean of these two differences. For bivariate distributions with equal marginals, that satisfy some additional conditions, a nonnegative r conveys the probability that the paired values of the two variables are identical by descent. This interpretation is inspired by the rationale of the genetic coefficient of inbreeding.