T-Test Calculator

A t-test checks whether a difference in means is large enough to be unlikely under the null hypothesis of no difference. The one-sample t-test compares a sample mean to a hypothesized population mean. The two-sample t-test compares the means of two independent groups. The result is a t statistic and degrees of freedom that you convert to a p-value using a t-distribution table or statistical software.

One-sample asks 'is this sample's mean different from a specific value (μ₀)?' Two-sample asks 'do these two groups have different means?' Use one-sample when you have a single group and a benchmark (a published average, a target, a historical value). Use two-sample when you have two independent groups (treatment and control, two product variants, two schools).

Welch's t-test does not assume equal variances between the two groups, which is a safer default than the classic 'pooled' two-sample t-test. When variances and sample sizes happen to be similar, Welch's test gives nearly identical results. When they differ, Welch's adjusts both the standard error and the degrees of freedom using the Welch-Satterthwaite approximation, which keeps the test calibrated.

t measures how many standard errors the observed difference is away from zero. Bigger |t| values mean stronger evidence against the null. The exact threshold depends on degrees of freedom and chosen significance level: for example, a two-tailed test at α = 0.05 with df = 30 needs |t| above about 2.04 to reject the null. Check a t-table or convert to a p-value to evaluate.

Computing a p-value from a t statistic requires integrating the t-distribution density, which is more involved than the simple closed-form math used elsewhere on the site. Rather than approximate it imperfectly, the calculator reports the t statistic and degrees of freedom, and you can drop them into a t-table, the p-value calculator on this site (for the z-approximation when df is large), or statistical software.

The t-test assumes the data are approximately normally distributed (especially important for small samples), the observations are independent within each group, and for the two-sample case, the two groups are independent. For very non-normal data, very small samples, or matched/paired data, other tests are more appropriate (Mann-Whitney U, Wilcoxon signed-rank, paired t-test). For formal research, confirm the right test with a qualified statistician.