Two-Sample t-Test Calculator (Pooled)
Compare the means of two independent groups with the pooled (equal-variance) Student t-test — get the t statistic, degrees of freedom, two-tailed p-value and a verdict.
How it works
t = (x̄₁ − x̄₂) ÷ √(s²ₚ·(1/n₁ + 1/n₂)) · s²ₚ = [(n₁−1)s₁² + (n₂−1)s₂²] ÷ (n₁+n₂−2) · df = n₁ + n₂ − 2
The pooled two-sample t-test asks whether two independent groups come from populations with the same mean, assuming their variances are equal. It merges the two sample variances into one pooled estimate, uses that to build the standard error of the difference, and compares the observed mean gap to that error. If the groups clearly have unequal variances, the Welch t-test on the main t-test page is the safer choice.
Worked example
Group 1 has mean 5.0 and group 2 has mean 5.62, five values each. The pooled standard deviation is about 0.19, giving a standard error near 0.12, a t of roughly −5.2 on 8 degrees of freedom, and a two-tailed p below 0.001 — the two means differ significantly.
Sources: Student’s t-test — Wikipedia
Frequently asked questions
When should I use the pooled test instead of Welch?
Use the pooled version only when the two groups have similar variances (a common rule of thumb is a ratio under about 3:1) and similar sizes. When variances differ, Welch’s unequal-variance t-test controls the error rate better and is the modern default.
What does the two-tailed p-value mean here?
It is the probability of seeing a mean difference at least this large in either direction if the two population means were actually equal. A p below your chosen α (often 0.05) is treated as evidence that the means differ.
Do the two groups need the same number of values?
No. The pooled test handles unequal group sizes; each variance is weighted by its own degrees of freedom. It does, however, assume the two populations share one common variance.
Related calculators
For education — confirm the test assumptions (independence, sample size, roughly normal data) before drawing conclusions from a p-value.