T-Test Calculator

One-sample t: tests H₀: μ = μ₀. t = (x̄ − μ₀) / (s/√n), df = n−1. Use when you have a single sample and a benchmark value.

Pooled two-sample t: tests H₀: μ₁ = μ₂ assuming equal variances. Uses the pooled SD sₚ. df = n₁ + n₂ − 2.

Welch's two-sample t: same hypothesis without assuming equal variances. SE = √(s₁²/n₁ + s₂²/n₂), df from the Satterthwaite approximation. Modern recommendation: default to Welch unless you've explicitly verified equal variances.

Independent observations within each group.

Approximately normal sampling distribution of the mean — automatic for n ≥ 30 by the CLT; check for skew/outliers in smaller samples.

Random sampling — or at least exchangeable conditions across groups.

Equal variances (pooled t only). Welch's t drops this assumption and is essentially always safe.

Manufacturer claims a battery lasts 22 hours. You test 10 units and get a mean of 23.4 hours, s = 1.7.

t = (23.4 − 22) / (1.7 / √10) = 1.4 / 0.5376 ≈ 2.604. df = 9.

Two-tailed p ≈ 0.029 — significant at α = 0.05. The sample provides evidence the true mean differs from 22 (likely higher).

The 95% CI on the mean is 23.4 ± t₀.₉₇₅,₉ · 0.5376 ≈ 23.4 ± 1.216 → (22.2, 24.6), consistent with rejecting 22.

The p-value is the probability, under H₀, of seeing a t at least as extreme as the one observed.

Small p (≤ α): reject H₀. The observed difference is unlikely under the null.

Large p: fail to reject H₀. The data don't refute the null — but they don't prove it either; absence of evidence is not evidence of absence.

Always report the effect size alongside p. A statistically significant difference of 0.2 hours of battery life is probably not practically meaningful.

Multiple testing: running 20 t-tests at α = 0.05 will produce one false positive on average. Use a correction (Bonferroni, Holm, FDR).

Using pooled t when the variances obviously differ. Welch is the safe default.

Paired data treated as independent. If each subject is measured twice, use a paired t-test (or one-sample t on the differences).

Reporting only p with no CI. The interval shows the magnitude of the effect, which is what decisions need.