What does the point-biserial correlation between a binary group flag and a continuous test score measure?

## It is Pearson's r wearing a hat When one variable is binary and the other continuous, you can still feed both into the ordinary Pearson correlation formula after coding the binary variable as 0 and 1. The result has a name, point-biserial r, but it is numerically identical to Pearson's r on that coding. Nothing new is being estimated; the name just signals the special structure, which happens to admit a much more interpretable algebraic form. ## The interpretable form `r_pb = ((M1 - M0) / s) * sqrt(p * (1 - p))` Here M1 is the mean score of the group coded 1, M0 the mean of the group coded 0, s the standard deviation of all n scores pooled together (computed with the divide-by-n convention), p = n1/n the share in group 1, and 1 - p the share in group 0. The first factor is a standardised mean difference: how far apart the groups sit, in units of overall score spread. The second is a balance factor. ## The balance factor is the part people forget sqrt(p(1-p)) is maximised at p = 0.5, where it equals 0.5, and it collapses toward 0 as either group becomes rare. Concretely, with a mean gap of one full standard deviation: a 50/50 split gives r_pb = 1.0 x 0.5 = 0.50, a 10/90 split gives 1.0 x 0.30 = 0.30, and a 1/99 split gives 1.0 x 0.0995 = about 0.10. The groups differ by exactly as much in all three cases, yet the correlation drops fivefold. This is why point-biserial r is a poor way to compare effects across datasets with different prevalence, and why a small r on a rare-group flag does not mean the groups are similar. The consequence runs the other way too: point-biserial r cannot reach 1 unless the split is balanced and the separation is complete. With a rare flag it is structurally capped well below 1, so judging it against the usual correlation rules of thumb understates the group difference. ## Sign is a coding artifact Swapping the labels, coding the other group as 1, flips the sign and leaves the magnitude untouched. Any linear recode does the same: using 1 and 2 instead of 0 and 1 changes nothing at all, since correlation is invariant to linear rescaling. So the sign carries meaning only once you say which group wears the 1, and any report of a point-biserial value must state that. ## Relationship to the two-sample comparison The same two numbers, the mean gap and the group sizes, drive both point-biserial r and the equal-variance two-sample t statistic on the same data. They are linked by `r_pb = t / sqrt(t^2 + n - 2)`, so knowing one gives the other. That identity is worth remembering because it explains a common confusion: the t statistic keeps growing as the sample grows for a fixed real gap, while r_pb stays put. One measures how strong the pattern is, the other how much data stands behind it, and inflating the sample changes only the second. ## Assumptions and traps Because it is Pearson's r, everything that troubles Pearson's r troubles this too: it is a summary of means and variances and is sensitive to outliers in the continuous variable, and it can only see a shift in level between the two groups. Two groups with identical means but very different spreads produce r_pb near 0 while being obviously different distributions, so plot the two distributions rather than trusting the coefficient alone. A related coefficient, the biserial correlation, is used when the binary variable is a dichotomised version of something underlying and continuous, such as pass/fail cut from an actual score. It answers a different question, estimating the correlation with the latent continuous variable, and generally comes out larger than the point-biserial value on the same data. Use point-biserial when the two groups are genuinely two categories, not a chopped continuum.