t-Test Calculator

One-sample: is the mean of a sample different from a stated value μ₀? t = (x̄ − μ₀) / (s/√n) with n − 1 degrees of freedom. Paired: two measurements on the same units (before/after) — the differences are taken and the one-sample test is run on them. Two-sample: two independent groups. With Welch's version the standard error is √(s₁²/n₁ + s₂²/n₂) and the degrees of freedom come from the Welch–Satterthwaite formula, usually not a whole number; the pooled version assumes equal variances and uses n₁ + n₂ − 2. Welch is the default because it costs almost nothing when the variances are equal and is right when they are not.

The p-value is the probability, if the true difference were zero, of a t at least as extreme as the one observed; it is computed from the t distribution (regularised incomplete beta), not read from a table. Two-sided is the default; choose one-sided only when the direction was fixed before the data were seen. “Significant” just means p is below the α you set. The confidence interval for the mean (or the difference) says how large the effect could plausibly be, and Cohen's d — the difference in pooled standard-deviation units — says how large it is regardless of sample size.

The t-test assumes independent observations and, for small samples, a population that is roughly normal; the page flags n below 30. It does not test more than two groups, proportions (see A/B Test) or non-parametric alternatives. Values are split on spaces, commas, semicolons and line breaks; tokens that are not numbers are ignored and listed. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.