How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

And this is Pearson’s formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors. If you want to get all trigonometric about it, the correlation is the cosine of the angle. It doesn’t matter if you remember what cosine means; you just need to know that the cosine of an angle is 1 when the angle is 0 (i.e., when the two vectors are pointing in the same direction) and −1 when the angle is 180 degrees (vectors pointing in opposite directions). Two variables are positively correlated when the corresponding vectors are separated by an acute angle—that is, an angle smaller than 90 degrees—and negatively correlated when the angle between the vectors is larger than 90 degrees, or obtuse.