T-Test Calculator
Run a one-sample, independent-samples or paired t-test.
How the t-test works
The t-test compares a sample mean against a known value (one-sample), two independent group means (independent-samples), or two related measurements (paired). The t statistic measures how far the observed difference is from what chance alone would produce, in units of the standard error.
For independent samples this calculator offers Welch's t-test as the default — it does not assume equal variances and is robust to unequal group sizes — alongside Student's t-test, which pools the variances and assumes them equal.
The result is a p-value read from Student's t distribution with the appropriate degrees of freedom. A p-value below your significance level is conventionally called statistically significant, but significance does not tell you the size or practical importance of the effect.
The formulas
One-sample: t = (x̄ − μ₀) / (s/√n) · Welch: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
For one sample, t compares the sample mean to the hypothesized value using the sample standard deviation. For independent samples, Welch's t divides the mean difference by the unequal-variances standard error, with degrees of freedom from the Welch–Satterthwaite formula. The paired t-test applies the one-sample formula to the within-pair differences.
Worked example
Two groups of three values
Group A = 1, 2, 3 and Group B = 4, 5, 6 give a mean difference of −3. Welch's t is −3.674 with 4 degrees of freedom and a two-tailed p-value of about 0.021 — significant at the 5% level.
Frequently asked questions
Which t-test should I use?
For two independent groups use Welch's t-test by default — it does not assume equal variances. Use Student's t-test only when equal variances are justified. For related measurements use the paired t-test.
What assumptions does the t-test make?
The data (or the differences, for a paired test) should be approximately normally distributed, or the sample large enough for the central limit theorem to apply. Observations should be independent. Entering data here does not prove normality — check it separately.
What is the difference between one-tailed and two-tailed?
A two-tailed test checks for a difference in either direction; a one-tailed test checks only for an increase or only for a decrease. Two-tailed is the common default.
Related tools
Related guides
- How to Choose the Right Statistical TestA practical framework for picking the right statistical test: start from your research question, data type and study design, then check the assumptions.
- Understanding P-Values and Statistical SignificanceWhat a p-value actually means, what it does not mean, the role of the significance level, and why the field is moving away from the 0.05 bright line.
- T-Test vs Z-Test vs ANOVAWhen to use a z-test, a t-test or an ANOVA, the assumptions behind each, and how the tests relate — including why F equals t² for two groups.
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