T-Test Calculator - One-Sample, Paired & Welch

Three options, picked by your data structure. Use a one-sample t-test when comparing one group's mean against a known or hypothesised value (does this batch's average weight match 500 g?). Use an independent two-sample t-test when comparing the means of two separate groups (test scores: boys vs girls). Use a paired t-test when measurements are linked — same subjects, two conditions (blood pressure before and after a drug). Pick wrong, and your p-value misleads. The decision tree is small but matters: independence, sample count, and pairing structure.

A paired t-test compares two measurements from the same subjects — like blood pressure before vs after treatment for the same patients. Each pair forms a difference, and the test asks whether the mean difference equals zero. An unpaired (independent) t-test compares means from two separate groups — say, group A receives a placebo and group B receives the drug, with different people in each group. Pairing reduces noise from individual differences, which often makes paired tests more powerful when the design supports it.

Welch's t-test compares two independent group means without assuming equal variances. The classic Student's t-test assumes both groups have the same variance — Welch's relaxes that assumption by adjusting both the standard error and the degrees of freedom. It's the safer default, especially when sample sizes are unequal or one group's variance looks much bigger than the other's. Modern software (R's t.test, Python's scipy) often uses Welch's by default for good reason. With equal variances and equal n, Welch's gives nearly identical results to Student's anyway.

The general form is t = (mean difference)/standard error, but the specifics depend on the test type. One-sample: t = (x̄ − μ0)/(s/√n). Independent two-sample (equal variances): t = (x̄1 − x̄2)/(sp × √(1/n1 + 1/n2)), where sp is the pooled SD. Paired: compute differences first, then t = d̄/(sd/√n). The structure is always the same — observed difference divided by how much variation you'd expect from random sampling alone. Software handles the pooling and df computation.

The p-value is the area in the t distribution beyond your computed t, with the appropriate tail. For a two-tailed test, you take the area in both tails (above |t| and below −|t|). Example: t = 2.10, df = 15. In Excel, =T.DIST.2T(2.10, 15) returns about 0.053. For a one-tailed test, =T.DIST.RT(2.10, 15) gives roughly 0.026. Bigger |t| or smaller df both shift toward smaller p-values. Always confirm whether your alternative hypothesis is one-sided or two-sided before reading the output.

Use it when you want to compare your single sample's mean against a known or hypothesised value. Examples: "Is the average waiting time at this clinic really 15 minutes, as advertised?" or "Does this batch of products meet the spec mean of 500 g?" You need one continuous variable, an approximately normal distribution (or a large enough sample for the CLT to apply), and independent observations. The test statistic compares your sample mean to the hypothesised value, scaled by the standard error.

Look at three things together. First, the p-value: if below your alpha (typically 0.05), the difference between the two group means is statistically significant. Second, the confidence interval for the mean difference: this shows the size and direction of the effect, not just whether it exists. Third, the effect size (Cohen's d) for practical importance. A significant p with a tiny mean difference may not matter in practice, while a non-significant result with a wide CI just means your sample is too small to draw firm conclusions.

Degrees of freedom (df) represent the number of independent pieces of information used to estimate variability. For a one-sample t-test, df = n − 1, because once the sample mean is computed, only n − 1 deviations are free to vary. For an independent two-sample test with equal variances, df = n1 + n2 − 2. Higher df makes the t distribution closer to the standard normal, making critical values smaller. For Welch's t-test, df is calculated using a more complex formula that accounts for unequal variances.