Matched Pairs

A matched-pairs design is simply one where every measurement in one group has a specific partner in the other: the same patient before and after, two twins, the same part on two machines. Once you subtract within each pair, you have one list of differences, and the formula is just the one-sample t-test you already know: $\bar d$ (the average difference) over $s_d/\sqrt n$ (its standard error), with $n$ (the number of pairs). Nothing new to learn, just a smarter way to set up the data.

Procedure: compute $d = \text{after} - \text{before}$ (or any consistent order) for each pair, then run a one-sample t-test on the $d$ values with $H_0: \mu_d = 0$.

Why pairing helps. People differ a lot from each other; each person differs less from themselves. Think about blood pressure: one person might sit at 140 and another at 110, a 30-point gap that has nothing to do with the diet. But if each drops by about 7, the differences are all close to 7. Subtracting removes the between-person variation, so real effects stand out with fewer subjects. There's a picture further down that shows this happening as the people spread apart.

A common mistake. It's tempting to run a two-sample test on before-and-after data, treating "before" and "after" as two groups. That throws away the pairing, and the person-to-person spread swamps the effect. If the same individuals are measured twice, subtract first and test the differences.

Before-and-after drug studies, twin studies, and comparing two measuring instruments on the same samples all use paired differences.