Pairing: subtract first, test second
When the same units are measured twice (before and after, left and right, matched twins), the paired t-test works on the differences: one number per pair, then an ordinary one-sample t-test of those differences against zero. The design's power comes from what the subtraction removes. People differ enormously in level; if strong students score high both times, the level differences swamp the change in the raw columns, but the differences column holds only the change itself, person by person.
Eight students, before: 72, 75, 80, 77, 74, 79, 73, 76; after: 78, 82, 85, 90, 88, 84, 79, 91. Differences: 6, 7, 5, 13, 14, 5, 6, 15, with mean 8.875 and SD 4.324. The standard error is 4.324 ÷ √8 = 1.529, so t = 8.875 ÷ 1.529 = 5.805 on 7 df, p = 0.0007. The 95% interval for the mean gain is 5.26 to 12.49 points. Run the same numbers as two independent samples instead and the standard error fattens from 1.529 to 1.979, t drops from 5.805 to 4.484, and the interval widens from 5.26–12.49 to 4.53–13.22: pairing is worth real evidence, because the person-to-person level differences never enter the differences column.
What pairing does not fix
Pairing removes level differences; it does not create a control group. The before-after gain here could be the course, or practice with the test format, or eight weeks of ordinary growth: the design cannot separate them, and the t-test only vouches for "the mean difference is not zero". Order effects and carryover are the paired design's other honest limits. Report the interval, name the rival explanations, and let the claim be sized to the design. The paired t-test calculator runs the numbers; the paired CI calculator isolates the interval.
Pairing things that are not actually linked. Arbitrarily matching rows from two independent groups and differencing them fabricates a design that does not exist and produces p-values with no meaning. Pairs must be the same unit twice, or deliberately matched before measurement.
Frequently asked questions
How is a paired t-test calculated?
Differences first, then a one-sample t-test on them: t = mean(d) / (SD(d)/sqrt(n)) with n - 1 degrees of freedom, n being the number of pairs.
Why is the paired test more powerful here?
The differences strip out each person's baseline. In the example the SD of differences is 4.324 against about 6 for the raw scores, so the same mean change is measured against less noise.
What if some pairs are incomplete?
Standard practice drops the pair from a paired analysis (the remaining difference is unobserved). If incompleteness is common or patterned, that itself is a finding to report.