This dichotomy is artificial. Notation is r with b subscript. Phi is when each of the varibles are naturally occuring, dichtomies. Ex. - gender with handedness. Notation is a circle with a line through it vertically.
Tetrachoric correlation is two artificially dichotomous variables.
They check to see if two samples come from the same population. Looks at 2 sample means. If so, they have the same mean. So let's say we have experimental and control group. Suppose we have a new way of teaching statistics.