t-Tests: Independent and Paired

The t-test is the workhorse for comparing two means, and it comes in two main flavors. The independent (two-sample) t-test compares the mean outcome between two separate groups, for example treatment versus control blood pressure. The paired t-test compares two measurements on the same subjects (or matched pairs), for example before versus after treatment in the same patients. Choosing wrong is a frequent error: pairing is more powerful when the same people are measured twice, because it removes the variability between different people and focuses on the within-person change. Assumptions: the outcome is continuous and roughly normal (for the independent test, in each group; for the paired test, the differences should be roughly normal); observations are independent; and for the independent t-test the two groups should have similar variances. Welch's t-test relaxes the equal-variance requirement and is a safe default. When the data are skewed or ordinal, switch to the non-parametric equivalent: Mann-Whitney U for independent groups, Wilcoxon signed-rank for paired data. Reading the output: you get a mean difference, a t-statistic, degrees of freedom, a p-value, and ideally a 95% confidence interval for the difference. Focus on the mean difference and its CI, the effect size and precision, not the p-value alone. Decide one-sided versus two-sided in advance; two-sided is the default. Pitfalls: running many t-tests across subgroups inflates false positives (a multiple-comparisons problem); using an independent t-test on paired data throws away power; and ignoring unequal variances. For three or more groups, do not run multiple t-tests, use ANOVA instead. Before trusting a t-test, check its assumptions quickly: a histogram of each group (or of the paired differences) for rough normality, and similar variances for the independent test, where Welch's version is the safe default when in doubt. And always report the mean difference with its 95% confidence interval, not merely whether p crossed 0.05, so readers can judge whether the effect is clinically meaningful.