---
title: "Point-biserial correlation"
method_id: SM.REL.COR.BIN.PTBIS
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/point_biserial_correlation.md
source: Clarus method library
---

# Point-biserial correlation

> A classical statistical method in the Clarus library (Relationship family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** point-biserial r; r_pb; rpb.

## Hypotheses

- **Null (H0):** rho_pb = 0
- **Alternative (H1):** rho_pb != 0

## When to use it

Use this method when your goal is to:
- quantify association continuous with binary
- measure relationship two variables

## Data it expects

- **dependent:** 1 continuous
- **independent:** 1 binary (dichotomous, 2 levels)
- **pairing:** paired_observations_same_unit

## Assumptions Clarus checks

- **normality group0** (gate: shapiro_wilk; severity: warning; on violation: warn_only)
- **normality group1** (gate: shapiro_wilk; severity: warning; on violation: warn_only)

## Effect size reported

- **r pb** (Cohen (1988))

## Honest limitations

- The sign of r_pb depends entirely on which category was coded 1 — always state the coding when reporting direction.
- Point-biserial r is restricted in range when the two groups are very unequal in size; an extreme split mechanically shrinks the maximum attainable |r|.
- If the binary variable was made by dichotomizing a continuous one, point-biserial r underestimates the true relationship — use biserial correlation.
- A significant correlation between a naturally-occurring group and an outcome is not evidence of causation.
- Results generalize only to the sampled population.

## Primary sources

- Tate, R. F. (1954). Correlation between a discrete and a continuous variable. Point-biserial correlation. The Annals of Mathematical Statistics, 25(3), 603-607. https://doi.org/10.1214/aoms/1177728730
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Point-biserial correlation", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/point_biserial_correlation.md

**APA**

Clarus. (2026). Point-biserial correlation (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/point_biserial_correlation.md

**BibTeX**

```bibtex
@misc{clarus-point-biserial-correlation,
  author       = {Clarus},
  title        = {Point-biserial correlation},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/point_biserial_correlation.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.REL.COR.BIN.PTBIS · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library