# Applying a Nonparametric Test to Paired Data

A parametric test assumes a specific distribution such as the normal distribution, and the t test is an example. A nonparametric test does not assume a particular distribution for the data. Most nonparametric tests are based on ranks and not the actual data values (this frees them from assuming a particular distribution). The study of nonparametric statistics can fi ll an entire textbook. We’ll just cover the high points and show how to apply a nonparametric test to your data.

The Wilcoxon Signed Rank Test The nonparametric counterpart to the t test is the Wilcoxon Signed Rank test. In the Wilcoxon Signed Rank test, we rank the absolute values of the original data from smallest to largest, and then each rank is multiplied by the sign of the original value (21, 0, or 1). In case of a tie, we assign an average rank to the tied values.

Table 6-4 shows the values of a variable, along with the values of the signed ranks. Figure 6-15 Normal probability plot of the Diff data Chapter 6 Statistical Inference 251 Table 6-4 Signed Ranks Variable Values Signed Ranks 18 7.0 4 2.0 15 6.0 25 23.5 22 21.0 10 5.0 5 3.5 There are seven values in this data set, so the ranks go from 1 (for lowest in absolute value) up to 7 (for the highest in absolute value).

### Final Words:

The lowest in absolute value is 22, so that observation gets the rank 1 and then is multiplied by the sign of the observation to get the sign rank value 21. The value 4 gets the sign rank value 2 and so forth. Two observations, 25 and 5, are tied with the same absolute value. They should get ranks 3 and 4 in our data set, but because they’re tied, they both get an average rank of 3.5 (or 23.5). Next we calculate the sum of the signed ranks.