---
title: "Pearson product-moment correlation"
method_id: SM.REL.COR.BIV.PEARSON
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/pearson_correlation.md
source: Clarus method library
---

# Pearson product-moment 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:** Pearson r; product-moment correlation; bivariate linear correlation.

## Hypotheses

- **Null (H0):** rho = 0 (no linear association in the population)
- **Alternative (H1):** rho != 0 (a linear association exists)

## When to use it

Use this method when your goal is to:
- quantify linear relationship two continuous
- measure strength direction association

## Data it expects

- **variables:** 2 continuous
- **pairing:** paired (each case has both measures)
- **privileged_dependent:** none (symmetric)

## Assumptions Clarus checks

- **linearity** (severity: warning; on violation: reroute)
- **bivariate normality** (gate: shapiro_wilk; severity: warning; on violation: reroute)
- **no influential outliers** (gate: bivariate_outlier_scan; severity: warning; on violation: warn_only)

## Effect size reported

- **correlation r** (Cohen (1988))

## Honest limitations

- Correlation is NOT causation: a relationship between {var_x} and {var_y} does not show that one causes the other; a lurking third variable may drive both.
- Pearson r measures only the STRAIGHT-LINE relationship — a strong curved (non-monotone) relationship can produce r near zero.
- r is sensitive to outliers and to restricted range: a single extreme point, or a truncated sample, can inflate or deflate r dramatically.
- Large n can make a trivially small r statistically significant — always read r and r^2 alongside p.
- Results generalize only to the sampled population and measurement range.

## Primary sources

- Pearson, K. (1896). Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia. Philosophical Transactions of the Royal Society A, 187, 253-318. https://doi.org/10.1098/rsta.1896.0007
- Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10(4), 507-521. https://doi.org/10.2307/2331838
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Ch.2.
- Field, A. (2013). Discovering Statistics Using IBM SPSS (4th ed.), Ch.7.
- Puth, M.-T., Neuhauser, M., & Ruxton, G. D. (2014). Effective use of Pearson's product-moment correlation coefficient. Animal Behaviour, 93, 183-189. https://doi.org/10.1016/j.anbehav.2014.05.003

## 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, "Pearson product-moment correlation", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/pearson_correlation.md

**APA**

Clarus. (2026). Pearson product-moment correlation (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/pearson_correlation.md

**BibTeX**

```bibtex
@misc{clarus-pearson-correlation,
  author       = {Clarus},
  title        = {Pearson product-moment correlation},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/pearson_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.BIV.PEARSON · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library