Wilcoxon Signed-Ranks Test for Matched Pairs

Use the Wilcoxon signed-ranks test when data does not follow a normal distribution or when testing matched pairs where normality cannot be assumed. This nonparametric test incorporates more information than the sign test, yielding more trustable conclusions. It requires matched pairs and approximately symmetric distribution of differences between pairs.

First, calculate the difference for each matched pair. Ignore signs and rank the absolute values in ascending order. Reapply original signs to create signed ranks. Sum positive and negative signed ranks separately. The test statistic T equals the smaller of these two sums, which is then compared to the critical value.

When sample size n is less than or equal to 30, use T as the test statistic, which is the smallest sum of signed ranks. For n greater than 30, calculate the Z test statistic using a specific formula. The sample size n represents the number of pairs with non-zero differences.

Two conditions are required: the data must consist of matched pairs, and the distribution of differences between these pairs should be approximately symmetric. These conditions ensure the test produces valid and reliable results for evaluating whether the median of differences equals zero.

If the test statistic T is less than the critical value obtained from the table, reject the null hypothesis, indicating significant differences between the paired measurements. If T is greater than or equal to the critical value, fail to reject the null hypothesis, suggesting no significant difference.

The Wilcoxon signed-ranks test incorporates more information than the sign test by using both the direction and magnitude of differences through ranking. This additional information allows the test to detect patterns and differences more effectively, resulting in more trustable conclusions about matched pair data.

Rejecting the null hypothesis means the median of the differences between matched pairs is not zero, indicating statistically significant differences between the two measurements. In the blood sugar example, rejecting the null hypothesis shows May and June levels are apparently not the same.