Correlation, Regression Lines, and Moments of Inertia

Abstract

A student of engineering or physics discussing a random variable X with mean μ and variance σ² might refer to μ as the center of gravity of the (probability) mass distribution of X, and to σ² as the moment of inertia of X about μ. Is there a similar “physical” interpretation of ρ, Pearson's product-moment correlation coefficient for pairs of random variables? We answer this question affirmatively by showing that ρ is equal to the ratio of a difference and sum of two moments of inertia about certain lines in the plane. From this observation it is easy to derive familiar important properties of ρ. Similar results hold for the population version of the nonparametric correlation coefficient Spearman's rho. These ideas are readily accessible by students in an undergraduate mathematical statistics course.

Key Words:

• Physics
• Spearman's rho
• Standard deviation lines