Point-biserial correlation
Association and correlation · reference distribution: t(n−2)
When to use it
Measure the relationship between a binary and a quantitative variable; it is equivalent to the two-sample t-test.
Null hypothesis
There is no correlation.
Assumptions
- Quantitative variable normal in each group
Test statistic
r_{pb} = \dfrac{\bar{x}_1 - \bar{x}_0}{s_n} \sqrt{\dfrac{n_1 n_0}{n^2}}, \qquad s_n = \sqrt{\tfrac{1}{n} \textstyle\sum (x_i - \bar{x})^2}Effect size
r_pb itself.
How to report it
r_pb(38) = .33, p = .038
In R and Python
R
cor.test(y, as.numeric(group == "B"))Python
stats.pointbiserialr(binary, y)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- Equivalent test: Two-sample t
Where it sits in the catalog
Association and correlation. Instead of comparing groups, they measure whether two variables move together, and how strongly.
In the decision tree
- What do you want to do? Measure the relationship between two variables
- What kind of variables are they? One binary and one numeric