One More Geometric Interpretation of Pearson’s Correlation

Abstract

It is known that Pearson’s correlation coefficient r is equal to the cosine of the angle between the vectors ⃗x and ⃗y describing the centered data. We use this known fact to show that r describes how closer the corresponding unit vector ⃗e_x =  is to the y-unit vector ⃗e_y =  than to its opposite −⃗e_y: namely, r is equal to the (scaled) difference between the squared distances d²(⃗e_x, −⃗e_y) and d²(⃗e_x, ⃗e_y). In particular, the correlation is positive if ⃗e_x is closer to ⃗e_y and negative if ⃗e_x is closer to −⃗e_y.