---
title: "Spearman rank-order correlation"
method_id: SM.REL.COR.BIV.SPEARMAN
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/spearman_correlation.md
source: Clarus method library
---

# Spearman rank-order 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:** Spearman rho; Spearman's rank correlation; rank-order correlation; Spearman r_s.

## Hypotheses

- **Null (H0):** rho_s = 0 (no monotonic association between the ranks in the population)
- **Alternative (H1):** rho_s != 0 (a monotonic association exists)

## When to use it

Use this method when your goal is to:
- measure monotonic association between two variables
- correlate two ordinal variables
- correlate two continuous variables robustly to outliers or nonlinearity

## Data it expects

- **dependent:** not_applicable (symmetric bivariate relationship)
- **independent:** 2 variables, each at least ordinal (ranks, ordinal scales, or continuous)
- **pairing:** paired (each case carries both measures)

## Assumptions Clarus checks

- **monotonicity** (severity: critical; on violation: warn_only)
- **paired observations** (severity: warning; on violation: correct)

## Effect size reported

- **correlation r** (Spearman (1904); Cohen (1988) bands applied to rho)

## Honest limitations

- Correlation does NOT imply causation — an unmeasured third variable may drive both.
- Spearman captures MONOTONIC association, not linear: a U-shaped relationship can produce rho near 0 despite a strong dependence — always inspect the scatterplot.
- rho-squared is the shared variance of the RANKS, not of the raw scores; do not interpret it as 'percent of variance explained' in the original units.
- Heavy ties make the p-value approximate; Kendall's tau-b is an alternative that handles ties more cleanly.
- Range restriction (a truncated sampling range) attenuates rho; the true relationship may be stronger.
- With very large n a negligible rho can be statistically significant.
- Results generalize only to the sampled population.

## Primary sources

- Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72-101. https://doi.org/10.2307/1412159
- 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.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch.7. 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, "Spearman rank-order correlation", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/spearman_correlation.md

**APA**

Clarus. (2026). Spearman rank-order correlation (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/spearman_correlation.md

**BibTeX**

```bibtex
@misc{clarus-spearman-correlation,
  author       = {Clarus},
  title        = {Spearman rank-order correlation},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/spearman_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.SPEARMAN · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library