The use of ranks releases statistical tests from the parametric assumptions about underlying distributions by replacing actual observed values with their rank within the ordered set of observed values. The Wilcoxon matched-pairs test is analogous to the matched-pairs t-test, but uses the sum of the ranks of the values associated with each of the two test subjects. The pairs are ordered according to the absolute values of their differences and then the sum of the ranks of the positive values is compared with the sum of the ranks of the negative values. If the two subjects exhibit no particular pattern in their relative behaviours then the positive and negative values should be distributed roughly evenly through the ranks and thus the rank-sums should be approximately equal. A distortion between the rank-sums indicates that one or other subject has a consistently superior performance over the other.
The test is defined as follows. Given a collection of pairs of data items, the differences between the pairs are found and ranked according to absolute magnitude. The sum of the ranks is then formed for the negative and positive differences separately. is the smaller of these two rank sums. For sufficiently large samples the following value is approximately normally distributed: