---
title: "Welch t-test (unequal variances)"
method_id: SM.DIF.PAR.2IND.WELCH
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/welch_t_test.md
source: Clarus method library
---

# Welch t-test (unequal variances)

> A classical statistical method in the Clarus library (Difference 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:** Welch's t-test; Welch-Satterthwaite t-test; unequal-variances t-test; Welch-Aspin test.

## Hypotheses

- **Null (H0):** mu_{group1} = mu_{group2}
- **Alternative (H1):** mu_{group1} != mu_{group2}

## When to use it

Use this method when your goal is to:
- compare two independent groups

## Data it expects

- **dependent:** 1 continuous (interval/ratio)
- **independent:** 1 nominal with exactly 2 levels (grouping)
- **pairing:** independent (between-subjects)
- **variance:** unequal or unknown between groups

## Assumptions Clarus checks

- **normality** (gate: shapiro_wilk; severity: warning; on violation: reroute)
- **independence** (severity: critical; on violation: warn_only)

## Effect size reported

- **cohen d** (Cohen (1988))

## Honest limitations

- Welch's test handles unequal variances but still assumes approximately normal data within each group; with small, skewed groups prefer Mann-Whitney U.
- Statistical significance with very large n may accompany a trivial effect size — report and read Cohen's d.
- A significant mean difference does not establish causation unless groups were formed by random assignment.
- Results generalize only to the population the sample was drawn from.

## Primary sources

- Welch, B. L. (1947). The generalization of 'Student's' problem when several different population variances are involved. Biometrika, 34(1-2), 28-35. https://doi.org/10.1093/biomet/34.1-2.28
- Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110-114. https://doi.org/10.2307/3002019
- Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92-101. https://doi.org/10.5334/irsp.82
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.).
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch.9.

## 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, "Welch t-test (unequal variances)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/welch_t_test.md

**APA**

Clarus. (2026). Welch t-test (unequal variances) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/welch_t_test.md

**BibTeX**

```bibtex
@misc{clarus-welch-t-test,
  author       = {Clarus},
  title        = {Welch t-test (unequal variances)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/welch_t_test.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.DIF.PAR.2IND.WELCH · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library