Differences

Information about the difference between the means of X and Y is contained in the values D = (Y - X) for each individual. The hypotheses

H0 :   μX = μY
HA :   μXμY

can then be expressed as

H0 :   μD = 0
HA :   μD ≠ 0

This reduces the paired data set to a univariate data set of differences, D, and reduces questions about (µY - µX) to questions about the mean of D.

Analysis of paired data

By taking differences between Y and X, much of the variability between the individuals is eliminated, making it easier to see whether their means are different. The example below shows paired data on the left with blue lines joining the x- and y-values in each pair. The differences on the right make it clearer that the y-values are usually higher than the corresponding x-values.