Point-Biserial and Phi Coefficients

Ordinary Pearson correlation is defined for two continuous variables, but plenty of practical questions pair a continuous measurement with a yes/no outcome — does a stock's daily return correlate with whether an earnings beat happened, coded as 0 or 1? The point-biserial correlation is exactly Pearson's correlation formula applied directly to that pairing, with the binary variable simply coded as 0 and 1; it can equivalently be computed as a scaled difference between the mean of the continuous variable in the "1" group versus the "0" group, which is why it is closely related to a two-sample $t$-test — a significant point-biserial correlation and a significant $t$-test on the same group difference will always agree.

The phi coefficient handles the case where both variables are binary — for instance, whether a trade was a buy (yes/no) and whether it occurred in the last hour of trading (yes/no) — and again is just Pearson correlation with both variables coded 0/1. It can be computed directly from a 2x2 contingency table's four cell counts, and it is mathematically identical to $\sqrt{\chi^2/n}$ for that table's chi-squared statistic, tying it directly to the standard test of independence for categorical data.