T-Test Calculator

Frequently Asked Questions

What is the Welch-Satterthwaite equation?

It approximates the degrees of freedom when the two samples have unequal variances: df = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]. The result is generally non-integer and less than n₁+n₂−2, giving a wider distribution and more conservative p-values.

When should I use a paired t-test?

Use paired when measurements are matched by design: before/after on the same subject, twins assigned to different treatments, left vs. right eye, or any setup where pairs share a common source of variation. Pairing removes that between-pair variation from the error term, increasing power substantially.

Why is Welch recommended over pooled by default?

Pooled gains minimal power when variances truly are equal, but it loses nominal control of the Type I error rate when variances differ - especially with unequal n. Welch performs well whether or not variances are equal. The safety of Welch outweighs the very small power advantage of pooled in equal-variance cases.

What does a non-significant result mean?

Failing to reject H₀ means you did not find convincing evidence against it given your sample size and variability. It does not prove H₀ is true. Equivalence tests or Bayesian analyses are better suited to establishing that an effect is negligibly small.

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How This Calculator Works

A t-test compares means when the population standard deviation is unknown and estimated from the sample. Three modes are available:

One-sample: tests H₀: μ = μ₀ with t = (x̄−μ₀)÷(s÷√n) and df = n−1.

Two-sample (Welch / pooled): compares two independent group means. Pooled assumes equal variances and uses df = n₁+n₂−2. Welch does not assume equal variances, computing df via the Welch-Satterthwaite equation.

Paired: reduces matched pairs to their difference scores and runs a one-sample test on those, with df = n−1.

The two-tailed p-value uses the Student's t CDF evaluated through the regularized incomplete beta function I_x(df/2, 1/2) via a Lentz continued fraction - accurate to about four decimal places.

Worked Example: One-Sample Test

A sample of n = 30 has mean x̄ = 52 and s = 6, tested against μ₀ = 50. SE = 6 ÷ √30 ≈ 1.0954. t = (52−50) ÷ 1.0954 ≈ 1.826 with df = 29. Two-tailed p ≈ 0.078. Against α = 0.05, this is not significant - we fail to reject H₀. The 2-point difference could be sampling noise given this spread and sample size.

Choosing the Right Test

One-sample: compare one group mean to a known or historical target value (μ₀).

Two-sample Welch (default): compare two independent groups. Safe default because it is robust to unequal variances and unequal sample sizes. Prefer this unless a formal equality-of-variance test (Levene's, Brown-Forsythe) supports the pooled version.

Two-sample pooled: only when group variances are genuinely similar and sample sizes are comparable. Gains a tiny amount of power over Welch in ideal conditions.

Paired: before/after measurements on the same subjects, or naturally matched pairs. Removes between-subject variation, dramatically increasing power for within-subject effects.

Assumptions and Common Mistakes

The t-test assumes the sampling distribution of the mean is approximately normal - reasonable for n ≥ 30 by the Central Limit Theorem, but fragile for small, heavily skewed samples. For n < 15 from a non-normal population, consider a non-parametric alternative like the Wilcoxon signed-rank or Mann-Whitney U test.

Using the pooled test when variances differ (especially with unequal n) inflates the false-positive rate. Treating paired data as independent throws away the pairing structure and loses power. Always enter sample standard deviations (n−1 denominator), not population standard deviations.

Interpretation and Effect Size

A significant t only says the observed mean difference is unlikely under H₀. It says nothing about how large or important the difference is. Always compute a confidence interval for the difference: for a one-sample t-test, the 95% CI for the true mean is x̄ ± t_0.025,df × s/√n.

Report Cohen's d = (m₁−m₂)/pooled SD as an effect size measure. With large samples, a trivially small difference can produce a highly significant t. With small samples, a practically important effect can fail to reach significance. The interaction between effect size, sample size, and significance is why studying both p and d together is essential for complete inference.

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