Welch's t-test (unequal variances)
Also known as: unequal variances t-test, Welch-Satterthwaite t-test, Welch t-Testi (Eşit Olmayan Varyans)
Welch's t-test is a parametric hypothesis test that compares the means of two independent groups without assuming their variances are equal. It was introduced by B. L. Welch in 1947 as a more robust generalization of Student's two-sample test for situations where the two groups have different spread.
Key highlights
- Does not assume equal variances, so it stays valid when the two groups have very different spread.
- More robust than Student's t-test and recommended whenever Levene's test is significant.
- In large samples it is insensitive to mild non-normality thanks to the central limit theorem.
- Reports an interpretable effect size (Cohen's d) and confidence interval alongside the p-value.
Intuition
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How it works
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When to use it
Use it to compare exactly two independent groups on a single continuous outcome when their variances are unequal — typically when Levene's test is significant (p < 0.05). It expects roughly normal data within each group (check with Shapiro-Wilk), though with large samples (the test sets a minimum around 20 observations) the central limit theorem makes it fairly insensitive to mild non-normality. With small samples and clear non-normality, the nonparametric Mann-Whitney U test is more appropriate.
Strengths & limitations
- Does not assume equal variances, so it stays valid when the two groups have very different spread.
- More robust than Student's t-test and recommended whenever Levene's test is significant.
- In large samples it is insensitive to mild non-normality thanks to the central limit theorem.
- Reports an interpretable effect size (Cohen's d) and confidence interval alongside the p-value.
- Still limited to comparing exactly two groups.
- Relies on approximate (Welch–Satterthwaite) degrees of freedom rather than an exact distribution.
- In small samples with clearly non-normal data it can be unreliable; a nonparametric test is then preferable.
Common pitfalls
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Applications
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Frequently asked
When should I prefer Welch over the Student t-test?
Use Welch whenever the two groups have unequal variances — in practice, when Levene's test is significant (p < 0.05) — or when group sizes are very different. Many statisticians recommend using Welch by default for two-group comparisons because it loses little power when variances happen to be equal but protects you when they are not.
Why are the degrees of freedom not a whole number?
Welch's test uses the Welch–Satterthwaite approximation, which combines the two separate variances and sample sizes into an effective degrees-of-freedom value. This blended quantity is usually fractional, which is normal and expected for this test.
Does Welch's test still need normality?
Each group's values should be approximately normal, checked with Shapiro-Wilk. However, with reasonably large samples the central limit theorem makes the test insensitive to mild departures from normality. With small samples and clear non-normality, switch to the Mann-Whitney U test.
What should I report alongside the result?
Report the mean difference, its 95% confidence interval, and Cohen's d as an effect size, in addition to the t value, the Welch degrees of freedom, and the p-value.
Sources
- 1.Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika, 34(1/2), 28–35.
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ScholarGate. (2026, June 1). Welch t-test. ScholarGate. https://scholargate.app/statistics/welch-t-test