Interpreting results: Wilcoxon signed rank test

The Wilcoxon signed rank test is a nonparametric test that compares the median of a column of numbers against a hypothetical median. Don't confuse it with the Wilcoxon matched pairs test which compares medians of two paired groups).

The nonparametric Wilcoxon signed rank test compares the median of a single column of numbers against a hypothetical median.

If the data were sampled from a population with a median equal to the hypothetical value you entered, what is the chance of randomly selecting N data points and finding a median as far (or further) from the hypothetical value as observed here?

If the P value is small, you can reject the idea that the difference is a due to chance and conclude instead that the population has a median distinct from the hypothetical value you entered.

If the P value is large, the data do not give you any reason to conclude that the population median differs from the hypothetical median. This is not the same as saying that the medians are the same. You just have no compelling evidence that they differ. If you have small samples, the Wilcoxon test has little power. In fact, if you have five or fewer values, the Wilcoxon test will always give a P value greater than 0.05, no matter how far the sample median is from the hypothetical median.

The Wilcoxon signed rank test does not assume that the data are sampled from a Gaussian distribution. However it does assume that the data are distributed symmetrically around the median. If the distribution is asymmetrical, the P value will not tell you much about whether the median is different than the hypothetical value.

Like all statistical tests, the Wilcoxon signed rank test assumes that the errors are independent. The term “error” refers to the difference between each value and the group median. The results of a Wilcoxon test only make sense when the scatter is random – that any factor that causes a value to be too high or too low affects only that one value.

2.	Ignore values that exactly equal the hypothetical value. Call the number of remaining values N.

3.	Rank these distances, paying no attention to whether the values are higher or lower than the hypothetical value.

4.	For each value that is lower than the hypothetical value, multiply the rank by negative 1.

7.	Add the two sums together. This is the sum of signed ranks, which Prism reports as W.

If the data really were sampled from a population with the hypothetical mean, you would expect W to be near zero. If W (the sum of signed ranks) is far from zero, the P value will be small.

With small samples, Prism computes an exact P value. With larger samples, Prism uses an approximation that is quite accurate.