Point-Biserial Correlation
Point-Biserial Correlation Coefficient · Also known as: rpb, r_pb, point biserial r, item-total correlation, dichotomous-continuous correlation
The point-biserial correlation coefficient (r_pb) measures the strength and direction of the linear association between one naturally dichotomous variable (coded 0/1) and one continuous variable. It is a special case of the Pearson product-moment correlation formally derived by Tate (1954) in the Annals of Mathematical Statistics and is the standard index used in psychometric item analysis, validity studies, and any research context where a binary grouping variable is related to a continuous outcome.
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When to use it
Use the point-biserial correlation when one variable is naturally and truly dichotomous (pass/fail, yes/no, present/absent — not an artificially cut continuous variable) and the other is continuous. Key assumptions: the continuous variable is at least approximately normally distributed within each group; observations are independent; the groups are mutually exclusive and exhaustive. When the dichotomy is artificial (an underlying continuous trait forced into two categories) use the biserial correlation instead, which corrects for the truncation. With very small samples (n < 20) the normal approximation is poor and bootstrap confidence intervals are preferable. The method is inappropriate when the continuous variable is severely skewed within groups or contains substantial outliers.
Strengths & limitations
- Directly interpretable as a standardized effect size on the familiar [−1, 1] scale.
- Algebraically identical to Pearson r, so it inherits all of Pearson's theoretical properties and is computed by any statistical package with no special function.
- The significance test is equivalent to the independent-samples t-test, providing a well-understood inferential framework.
- Standard and widely expected in psychometric test-item reports, enabling direct cross-study comparison.
- Sensitive to extreme imbalance between the two group sizes: when one group is very small the maximum attainable |r_pb| is substantially less than 1, making comparisons across items or studies with different p-values (item difficulty) misleading.
- Assumes the dichotomy is natural; applying it to an artificially dichotomized continuous variable underestimates the true association — use the biserial coefficient in that case.
- Like Pearson r, it is a measure of linear association only and can miss monotone non-linear relationships.
- Does not extend directly to polytomous (more than two category) grouping variables; use the polyserial or general Pearson correlation in that setting.
Frequently asked
Is the point-biserial correlation just the Pearson correlation with 0/1 codes?
Yes, exactly. If you enter the dichotomous variable as 0s and 1s and compute Pearson r with the continuous variable, you obtain r_pb. The dedicated formula is simply an algebraically equivalent rearrangement that makes the role of group means and group sizes explicit, which is useful for hand calculation and for understanding item-analysis output.
How is point-biserial r different from biserial r?
Point-biserial r treats the dichotomy as a genuine two-category variable. Biserial r assumes the binary split masks an underlying normally distributed latent trait and applies a correction factor (dividing by the ordinate of the standard normal at the cut point). Biserial r is always larger in absolute value than point-biserial r from the same data and can even exceed 1 in magnitude with extreme splits. Use point-biserial for truly natural dichotomies and biserial for artificially dichotomized continuous variables.
What is an acceptable r_pb value for a test item?
In classical test theory a common rule of thumb flags items with r_pb below 0.20 as poor discriminators, though thresholds vary by context and test purpose. Values of 0.30 and above are generally considered good. Negative values indicate an item that high-ability examinees fail more often than low-ability ones — such items should be reviewed for scoring errors or content problems.
Does a significant p-value prove the item is good?
No. Statistical significance in large samples can accompany trivially small r_pb values. Conversely, in very small item-tryout samples (n < 100) a genuinely discriminating item may not reach significance. Always interpret r_pb in terms of its magnitude and confidence interval, not solely by the p-value.
Sources
- Tate, R. F. (1954). Correlation between a discrete and a continuous variable. Point-biserial correlation. Annals of Mathematical Statistics, 25(3), 603–607. DOI: 10.1214/aoms/1177728730 ↗
- Tate, R. F. (1955). The theory of correlation between two continuous variables when one is dichotomized. Biometrika, 42(1–2), 205–216. DOI: 10.1093/biomet/42.1-2.205 ↗
- Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric Theory (3rd ed.). McGraw-Hill. ISBN: 978-0070478497
How to cite this page
ScholarGate. (2026, June 3). Point-Biserial Correlation Coefficient. ScholarGate. https://scholargate.app/en/statistics/point-biserial-correlation
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Independent t-testStatistics↔ compare
- Item Response TheoryPsychometrics↔ compare
- Pearson CorrelationStatistics↔ compare
- Spearman CorrelationStatistics↔ compare