Solved – Proof of Point-Biserial Correlation being a special case of Pearson Correlation

Let the $n$ data consist of $n_0gt 0$ $(x, 0)$ pairs and $n_1gt 0$ $(x, 1)$ pairs. Their Pearson correlation coefficient will be the same as the reversed data consisting of corresponding $(0,x)$ and $(1,x)$ pairs. Because there are exactly two distinct values of the first coordinates, the regression line of the reversed data must pass through the mean points $(0,M_0)$ and $(1,M_1)$, whence it has slope $(M_1-M_0)/(1-0) = M_1-M_0$. The correlation coefficient is obtained by standardizing this: it must be multiplied by the standard deviation of the first coordinates and divided by the standard deviation of the second coordinates (the original $x$ values), written $s_n$. The standard deviation of the first coordinates is readily computed from the fact that they consist of $n_0$ zeros and $n_1$ ones; it equals

$$sqrt{frac{n_1}{n}left(1-frac{n_1}{n}right)} = sqrt{frac{n_0n_1}{n^2}}.$$

Consequently the Pearson correlation coefficient is

$$r = frac{M_1-M_0}{s_n}sqrt{frac{n_0n_1}{n^2}},$$

which is precisely the Wikipedia formula for the point-biserial coefficient.

The heights of the red dots depict the mean values $M_0$ and $M_1$ of each vertical strip of points. The dashed gray line is the regression line.