Point-Biserial Correlation — continuous vs. dichotomous

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Pearson correlation between a CONTINUOUS outcome y and a DICHOTOMOUS predictor x coded 0/1.

Mathematically identical to the independent-samples t-test re-expressed as a correlation: r_pb and Cohen's d encode the same effect. Reports r_pb, Fisher-z 95% CI, equivalent t and p, per-group descriptives, Cohen's d, AND the theoretical r_pb ceiling imposed by the observed binary split (max |r_pb| = 2·sqrt(b)/(1+b) where b is the balance ratio min(n0,n1)/max(n0,n1)).

Worked example

Does passing a course relate to weekly study hours?

For 70 students, pass/fail (binary) and weekly study hours (continuous) were recorded; the point-biserial correlation measures the link between a binary and a continuous variable.

Result

Study hours were associated with passing, r_pb = .65, p < .001 — those who passed had studied more.

How you'd report it (APA)

A point-biserial correlation showed study hours were significantly associated with passing, r_pb = .65, p < .001.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Treatment vs control on a continuous outcome
    A trial randomises 80 patients to drug vs placebo and measures systolic blood pressure at 12 weeks.
  • Naturally-binary demographic predictor
    An education researcher measures reading speed in 200 schoolchildren and computes r_pb between sex and reading speed.
  • Item-total correlation in psychometrics
    A 40-item knowledge test is administered to 500 students.

When NOT to — use instead

Hypotheses

H₀: ρ_pb = 0 — no linear relation between the binary grouping and the continuous outcome.
Hₐ: ρ_pb ≠ 0.

Parameter tested: point-biserial correlation ρ_pb

Assumptions (and what to do if they fail)

Equal variances in the two groups of x. A strong Levene's test violation inflates the SE of r_pb (parallel to Welch's adjustment in t-tests).medium

Check: Levene's p ≥ .05 and SD ratio ≤ 2.

If violated: Unequal variances inflate the Type I error of the equivalent t-test.

No extreme outliers in either group — |z| > 3 rule of thumb.medium

Check: Per-group |z| > 3 count.

If violated: A single outlier can drive r_pb up or down.

Groups are not extremely unbalanced. r_pb is bounded above by √(p·q)·(1/√(p·q+1)) ≈ 0.8 when p = 0.5, but only 0.4 when p = 0.05 — the ceiling shrinks with imbalance.low

Check: smaller_n / larger_n ≥ 0.3.

If violated: Severe imbalance caps the maximum achievable r_pb.

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