ttest

A t-test is a statistical hypothesis test used to determine whether there is a significant difference between the means of two groups. It relies on the t-distribution and was developed by William Sealy Gosset under the pseudonym Student. The test is commonly applied in small-sample scenarios and forms a core part of inferential statistics.

There are several variants of the t-test. The one-sample t-test compares a sample mean with a known value. The independent (two-sample) t-test compares the means of two independent groups. The paired t-test compares means from related or matched units, such as measurements taken on the same subjects before and after an intervention. The independent t-test has two main versions regarding variances: the standard version assumes equal variances, while the Welch version does not and provides an adjusted degrees of freedom.

The test statistic t is calculated as the difference between means divided by the standard error of the difference. Degrees of freedom depend on the version: one-sample and paired tests use n − 1, the independent test with equal variances uses n1 + n2 − 2, and the Welch version uses a refined, non-integer df. Assumptions include that data are approximately normally distributed, measured on at least an interval scale, and that observations are independent. The result is a p-value indicating the probability of observing the data under the null hypothesis of equal means. A small p-value leads to rejection of the null in favor of a difference between means. Reporting effect size, such as Cohen’s d, is common to convey practical significance.

Software implementations include R’s t.test, Python’s scipy.stats functions such as ttest_ind and ttest_rel, and similar routines in other statistical packages. Alternatives exist when normality or equal-variance assumptions are violated.