Home › Statistics & Probability Calculators › Two Sample T-Test Calculator (Welch, Pooled)
Two Sample T-Test Calculator
The independent two-sample t-test compares the means of two separate groups, such as test scores of two classes or reaction times under two conditions. Enter each group’s raw data or its mean, standard deviation and size, then choose Welch’s t-test (unequal variances, the safer default) or the pooled t-test (equal variances assumed).
The output shows both groups’ statistics, the difference of means, the standard error and degrees of freedom for the chosen method (including the Welch–Satterthwaite formula), the t statistic, the p-value, the critical value and a confidence interval for μ₁ − μ₂. Cohen’s d using the pooled standard deviation is reported as the effect size.
Welch’s version does not assume the two groups have the same variance and loses very little power when they do, which is why R, SciPy and most modern courses make it the default. Use the pooled version only when your course requires it or an F-test for equal variances supports it. If the two samples are matched pairs, use the paired t-test.
Separate values with commas, spaces or new lines. Leave blank to use the summary boxes.
Separate values with commas, spaces or new lines. Leave blank to use the summary boxes.
Result and step-by-step solution
- State the hypothesesH₀: μ₁ − μ₂ = 0 H₁: μ₁ − μ₂ ≠ 0 (two-tailed test)
- Sample statisticsGroup 1 (n = 8): x̄₁ = Σx ÷ n = 674 ÷ 8 = 84.25; s = √[Σ(x − x̄₁)² ÷ (n − 1)] = √(225.5 ÷ 7) = 5.6758.
Group 2 (n = 9): x̄₂ = Σx ÷ n = 680 ÷ 9 = 75.5556; s = √[Σ(x − x̄₂)² ÷ (n − 1)] = √(226.2222 ÷ 8) = 5.3177. - Difference of meansx̄₁ − x̄₂ = 84.25 − 75.5556 = 8.6944
- Standard error and degrees of freedomSE = √(s₁²/n₁ + s₂²/n₂) = √(32.2143/8 + 28.2778/9) = 2.6775. Welch–Satterthwaite df = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)] = 14.475.
- Test statistict = (x̄₁ − x̄₂) ÷ SE = 8.6944 ÷ 2.6775 = 3.2473
- p-value2 × P(T ≥ |3.2473|) = 0.0056 (t distribution, df = 14.47)
- Decisionp = 0.0056 < α = 0.05. critical values ±2.1382 (reject when |t| > 2.1382).
| 95% CI for μ₁ − μ₂ | 2.9695 to 14.4194 |
|---|---|
| Cohen’s d (pooled SD) | 1.584 |
Frequently asked questions
- Should I use Welch's or the pooled t-test?
- Welch’s, unless you have a specific reason to assume equal variances. It is valid whether or not the variances are equal and gives nearly identical results when they are.
- How are the Welch degrees of freedom calculated?
- df = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]. The value is usually not a whole number; the calculator uses it exactly rather than rounding down.
- Do the groups need the same size?
- No. Both versions handle unequal sample sizes. Very unequal sizes combined with unequal variances are exactly where Welch’s test matters most.
- What if my data are not normal?
- With moderate sample sizes the t-test is robust. For small, skewed samples or ordinal data use the Mann-Whitney U test instead.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.