ttests

T-tests are statistical methods used to determine whether there is a significant difference between the means of groups or between a group mean and a reference value. They are among the most common inferential tests in the behavioral, social, and biomedical sciences.

There are several variants. The one-sample t-test assesses whether the mean of a single sample differs from a hypothesized value. The independent two-sample t-test compares the means of two independent groups to see if they are different. The paired t-test compares means from the same group at two time points or from matched pairs, by analyzing the differences within pairs. The independent two-sample t-test can assume equal variances (Student’s t) or use Welch’s t, which does not assume equal variances.

The test statistic for a t-test is t = (x̄1 − x̄2) / SE, where x̄ are group means and SE is the standard error of the difference. Degrees of freedom depend on the variant: one-sample and paired t-tests have df = n − 1; the simple two-sample t-test with equal variances has df = n1 + n2 − 2. A p-value is obtained from the t-distribution, typically two-tailed unless a directional hypothesis is specified.

Assumptions include that the data are approximately normally distributed in the population, observations are independent (within groups for the independent test), and the data are measured on an interval or ratio scale. For independent t-tests, equal variances are assumed in the classic version; Welch’s t-test relaxes this. When assumptions are violated, nonparametric alternatives such as the Wilcoxon signed-rank test or Mann–Whitney U test may be used.