Last updated: 2026-06-23

What This Calculator Does

This calculator runs a two-sample t-test from summary statistics. Given the mean, standard deviation, and sample size of two independent groups, it reports the t statistic, the degrees of freedom, and a two-tailed p-value — the standard tools for deciding whether two group means differ by more than sampling noise alone.

What a two-sample t-test answers

Suppose two groups produce different averages — a treatment group and a control group, version A and version B, samples from two factories. The averages will almost never be identical, even when the underlying processes are the same, simply because every sample carries random variation. The two-sample t-test asks a precise question: if the two populations truly had equal means, how surprising would a difference this large be? The answer is the p-value.

Welch's t formula

This calculator uses Welch's t-test, which does not assume the two groups share a common variance. The t statistic is the difference in means divided by the standard error of that difference:

where $\bar{x}_1, \bar{x}_2$ are the sample means, $s_1, s_2$ the sample standard deviations, and $n_1, n_2$ the sample sizes. The denominator is the standard error — it grows with the groups' variability and shrinks as samples get larger, so the same difference in means becomes more convincing with more data.

Why Welch over the pooled (Student's) test

The classic Student's t-test pools both standard deviations into a single variance estimate, which is only correct when the two populations have equal variance. Welch's version keeps the two variances separate, as the formula above shows. Because equal variances are the exception rather than the rule in real data, Welch's test is the recommended default. It performs almost identically to the pooled test when variances happen to match, and it stays accurate when they do not — particularly when the groups also differ in size.

Degrees of freedom (Welch–Satterthwaite)

Because the variances are not pooled, the reference distribution does not have a simple integer degrees of freedom. Welch's test approximates it with the Welch–Satterthwaite equation:

The result is generally not a whole number, and it lies between the smaller of $n_1 - 1$ and $n_2 - 1$ and their sum. Fewer degrees of freedom give the t-distribution heavier tails, which makes the test appropriately more cautious with small or unequal samples.

The p-value and how to read it

Once $t$ and $df$ are known, the two-tailed p-value is

where $F_t$ is the cumulative distribution function of the t-distribution. The factor of two and the absolute value count departures in both directions — a larger group 1 mean or a larger group 2 mean.

Compare the p-value to a significance level $\alpha$ chosen before looking at the data, conventionally 0.05:

• p < α — the difference is statistically significant; the data give evidence against equal means.
• p ≥ α — the data do not provide enough evidence to reject equal means (which is not the same as proving them equal).

The p-value is not the probability that the means are equal, and a non-significant result is not proof of "no difference." It only measures how compatible the data are with the equal-means assumption.

Worked example

Group 1 has mean 10, standard deviation 2, and 30 observations; group 2 has mean 12, standard deviation 2.5, and 30 observations.

With $p \approx 0.0012$, well below 0.05, the difference between the two group means is statistically significant.

A significant p-value tells you a difference is detectable, not how large it is. Pair it with an effect-size measure such as Cohen's d Calculator, which expresses the gap between means in standard-deviation units and does not inflate with sample size.

Assumptions

The two-sample t-test rests on a few conditions:

• Independence — the two groups are independent, and observations within each group are independent. Paired or repeated measurements need a paired test instead.
• Approximate normality — each group is roughly normally distributed, or the samples are large enough that the central limit theorem makes the sampling distribution of the mean approximately normal. With moderate sample sizes and no extreme skew, the test is fairly robust.
• Unequal variances allowed — Welch's form handles different group variances directly, so you do not need to test for or assume equal spread.

Prefer Welch's t-test as the routine default for comparing two means. When your test statistic is already a standardized z rather than summary statistics, the P-Value from Z-Score Calculator converts it to a p-value using the normal distribution instead.