Matched pairs design

A matched pairs design is a randomized block design with blocks of size two: two treatments, and each block supplying exactly one observation of each. It comes in two forms: two units matched on a variable expected to affect the response, with the two treatments randomly assigned one to each member of the pair, or a single unit that takes both treatments with the order randomized. Either way the data collapse to one difference per pair, and $n$ is the number of pairs, not the number of measurements.

The site's paired t practice runs one data set both ways. Eight athletes have their vertical jump measured before and after a training program, giving differences of 3, 3, 5, 2, 3, 4, 1 and 5 centimeters, so $\bar{x}_d$ (x-bar-d) is 3.25 and $s_d$ is 1.3887. Then $SE = 0.4910$ and $t = 6.62$ on 7 degrees of freedom, a one-sided p-value near 0.00015. The same sixteen numbers treated as two independent samples give $SE = 2.2971$ and $t = 1.41$, with a p-value near 0.100. Only the standard error changed. The before and after columns correlate at 0.98, and that shared athlete-to-athlete variation is what the pairing removes.

"A paired design always gives a narrower interval" is false, and degrees of freedom are why. With 10 pairs the paired interval uses $t^*$ on 9 degrees of freedom while a two-sample analysis of the same 20 observations uses 18. If the two measurements inside a pair carry no correlation, the two standard errors come out algebraically identical, so the paired interval is about 7.7 percent wider for nothing. Pairing has to earn that back, and at 10 pairs the break-even within-pair correlation is roughly 0.14.

Equal group sizes never establish pairing. They only fail to rule it out, and the link has to come from how the data were produced.

Topic 1.13 names matched pairs as one of three designs, and its analysis is topic 4.5, a test for a population mean difference.