Welch's T-Test

Compare the means of two independent samples without assuming equal variances. Enter summary statistics and get Welch’s t, the Welch–Satterthwaite degrees of freedom and a two-tailed p-value.

How the calculation works

For each group you need the sample mean, the sample standard deviation (computed with n − 1) and the sample size n (at least 2). The calculator first finds the standard error of the difference, SE = √(s₁²/n₁ + s₂²/n₂), and then the test statistic t = (x̄₁ − x̄₂) / SE.

Degrees of freedom use the Welch–Satterthwaite equation: df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]. The result is usually fractional and smaller than n₁ + n₂ − 2. The two-tailed p-value comes from Student’s t distribution with that df, and the calculator flags whether the difference is significant at α = 0.05.

The test was introduced by B. L. Welch in 1947. Because it stays accurate when variances and group sizes differ, many statisticians recommend it as the default two-sample t-test. A positive t means group 1 has the larger mean.

Worked example

Group 1: mean 20, SD 5, n = 30; group 2: mean 17, SD 8, n = 25 (the defaults). SE = √(25/30 + 64/25) = √3.393 ≈ 1.842; t = 3 / 1.842 ≈ 1.629; df ≈ 38.8; p ≈ 0.111, so the difference is not significant at α = 0.05. The matching 95% confidence interval for the difference is 3 ± 2.023 × 1.842, roughly −0.7 to 6.7, which includes zero.

Things to keep in mind

• Enter standard deviations, not standard errors. If a paper reports SEM, multiply it by √n first.
• This test is for independent groups. Before/after measurements on the same people need a paired t-test.
• With small samples (n below about 15–20) and strongly skewed data, consider a Mann–Whitney test or a bootstrap instead.
• A non-significant result does not prove the means are equal — the sample may simply be too small to detect a real difference.
• Report the effect size too: the mean difference with its confidence interval is more informative than p alone.