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t-test calculator

Test a mean against a target, compare two means, or compare paired measurements. Enter summary statistics, or raw data for the paired test.

Sample 1
Sample 2

How it is calculated

One-sample: t = (x̄ − μ₀) / (s/√n) df = n − 1 Pooled: t = (x̄₁ − x̄₂) / (sp·√(1/n₁ + 1/n₂)) df = n₁ + n₂ − 2 Welch: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) df from the Welch–Satterthwaite formula Paired: one-sample test on the differences d = A − B, against 0 Example (Welch, the default above): t = 1.774, df = 21.3, p = 0.0903

Reject H₀ when p < α, or equally when |t| is beyond the critical value. Always report the confidence interval as well: it shows how big the difference is, which the p-value does not.

Read more: Which hypothesis test to use · Type I vs Type II error · Confidence intervals · Hypothesis testing in DMAIC

FAQ

Should I use a pooled or a Welch two-sample t-test?

Welch is the safer default: it does not assume the two groups have the same variance and loses very little when they do. Use pooled only when you have good reason to believe the variances are equal, or when an exam question tells you to assume it.

When is a paired t-test the right one?

When each value in one group is matched to one value in the other: the same part measured before and after, or the same sample measured by two gages. Pairing removes part-to-part variation, so it can find smaller differences.

What does the p-value mean here?

If the true difference were zero, the p-value is the chance of seeing a t statistic at least this far from zero. It is not the chance that the null hypothesis is true. See the p-value guide.

t-test or z-test?

Use z only when the population standard deviation is known. With a sample standard deviation, use t. For large samples the two give nearly the same answer.

What assumptions does the t-test make?

Independent observations and roughly normal data (or of the differences, for a paired test). With 30 or more per group the test tolerates moderate non-normality. For skewed small samples consider a nonparametric test.

Hypothesis tests are core exam material

This shows up on the exams as a calculation question. Practice it with new numbers every time, and every wrong answer explained.

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