pairedsample

Paired sample, or paired data, refers to a set of observations consisting of two measurements taken on the same experimental unit or on units that have been matched in pairs. Each pair links measurements that are naturally related—for example, a subject measured before and after an intervention, or two siblings matched on age and sex. The paired structure reduces within-unit variation, increasing sensitivity to detect differences between conditions.

In analysis, researchers typically compute the difference D_i = X_{1i} − X_{2i} for i = 1 to n. The primary goal is to test whether the mean difference is zero. For a paired t-test, the test statistic is t = mean(D) / (sd(D)/sqrt(n)), assuming the differences are approximately normally distributed. If the normality assumption is questionable, nonparametric alternatives such as the Wilcoxon signed-rank test or the sign test may be used.

Key assumptions include that the pairs are chosen independently and that the measurement scale is at least interval, with the difference scores being the focus of the analysis. The method is particularly suitable for before-and-after studies, pre-post comparisons, and matched-subject designs where each unit contributes two related observations.

Extensions of the idea include more complex repeated measures designs, where multiple related measurements are taken per subject. In such cases, methods like repeated-measures ANOVA or linear mixed-effects models may be employed to account for correlations within subjects and to handle additional time points or conditions.