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 research methods

Hypothesis Testing
ANOVA

When to use this tool

Use a t-test when you compare exactly two means: one sample against a known value, two independent groups, or two related measurements (paired).

Assumptions

Approximate normality of the data or differences (or a large enough sample for the central limit theorem).
Independent observations (except within pairs for a paired test).
For Student's t-test, approximately equal variances — otherwise use Welch's t-test (the default here).

Limitations

A significant result does not tell you the size or practical importance of the difference.
With many group comparisons the chance of a false positive rises; consider a correction.

How to interpret the result

The p-value is the probability of observing a difference at least this extreme if the null hypothesis were true. p < α is conventionally called statistically significant.

How to report the result

Report the test type (one-sample / independent / paired), the t statistic, degrees of freedom, p-value, and ideally the group means and an effect size — for example, 't(28) = 2.31, p = 0.028, d = 0.84'.

Next steps in your analysis

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