Welch's T-Test Calculator
Comparing Two Groups Without Assuming Equal Variance
The classic Student's t-test for two independent samples assumes both groups share the same underlying variance — an assumption that frequently doesn't hold in practice, especially when group sizes differ. Welch's t-test drops that requirement entirely. It's now widely recommended as the default two-sample t-test, since it performs about as well as Student's version when variances happen to be equal and considerably better when they aren't.
The Formula
df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1−1) + (s2²/n2)²/(n2−1)]
The df formula — the Welch-Satterthwaite equation — typically produces a non-integer value, unlike the simple n1+n2−2 used in the equal-variance version.
Worked Example
Group 1: 4, 8, 15, 16, 23. Group 2: 23, 42, 5, 9, 17.
| Quantity | Group 1 | Group 2 |
|---|---|---|
| Mean | 13.20 | 19.20 |
| Variance | 54.70 | 211.20 |
| Quantity | Value |
|---|---|
| t statistic | −0.8228 |
| df (Welch-Satterthwaite) | 5.94 |
| Two-tailed p-value | 0.4424 |
Computed directly from the calculator's formula. Note the fractional degrees of freedom, characteristic of Welch's correction.
Group 2's variance here is nearly four times Group 1's — exactly the kind of imbalance where using the standard pooled-variance t-test would understate the uncertainty and risk a false positive.
Where This Calculation Matters
- Unequal sample sizes — when one group has substantially more observations than the other, unequal variances become especially damaging to the standard t-test's accuracy.
- A/B testing — conversion or engagement metrics between two variants frequently have different spreads, making Welch's version the safer default in experimentation platforms.
- General best practice — many statisticians recommend using Welch's t-test by default for any two-sample comparison, reserving the equal-variance version only for cases where equal variance is confirmed.
How to Use This Calculator
- Enter Group 1 values as a comma-separated list.
- Enter Group 2 values as a comma-separated list (each group needs at least 2 values).
- Select Calculate to get the t statistic, Welch-Satterthwaite degrees of freedom, and two-tailed p-value.
Related Calculations
For paired (dependent) samples, use the Paired T-Test Calculator. When equal variances can be assumed, the standard T-Test Calculator applies.