Does the Correlation Coefficient Really Measure the Degree of Clustering Around a Line?

Abstract

This paper derives a formula for the square of the correlation coefficient r² in terms of its maximum possible value under rotation of the axes. The value of r² is minimized at 0 when the principal components of the scatterplot coincide with the axes, and is maximized when the principal components are at ±45° to the axes. An application of the formula to quick visual estimation of r is indicated.