Welch's T-Test Calculator
Compare two group means without assuming equal variances — the test that should be the default for two independent samples.
Welch's t uses each group's own variance: t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂), with the Welch–Satterthwaite degrees of freedom, which are usually fractional.
How the welch's t-test calculator works
Welch's t uses each group's own variance: t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂), with the Welch–Satterthwaite degrees of freedom, which are usually fractional.
It costs almost nothing when the variances really are equal and saves you when they are not, which is why modern practice recommends it over the pooled Student's t-test as the routine choice.
Formula: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Worked examples
| Inputs | t statistic | Note |
|---|---|---|
| Means 52 and 47, unequal spread | 1.583251 | t = 1.57, p = 0.126 |
| Identical groups | 0 | t = 0, p = 1 |
| A large clear difference | 4.749754 | strongly significant |
FAQFrequently asked questions
Why use Welch rather than Student?
Because it does not assume the two groups share a variance, and it loses almost nothing when they do. Statisticians increasingly recommend it as the default.
Why are the degrees of freedom fractional?
The Welch–Satterthwaite approximation interpolates between the two groups' information. A non-integer df is normal and correct.
What if my group sizes are very different?
That is exactly where pooling goes wrong. The pooled test can be badly anti-conservative when the smaller group has the larger variance.
Do I need to test for equal variances first?
No — and doing so is worse. A preliminary variance test inflates the overall error rate. Just use Welch.
What does the confidence interval add?
The size of the difference and its precision. A p-value alone tells you only whether zero is plausible.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.