The Wilcoxon test is a widely used nonparametric test that assesses whether two paired sets of differences are statistically significant. It is most appropriate when the data samples are related or paired, such as comparing pre-test and post-test scores for the same group of individuals or for comparing stock price changes over two different time periods. This test does not make assumptions about the underlying distribution of the data and is suitable for small sample sizes.

The Wilcoxon test comes in two versions, the rank sum test and the signed rank test. The rank sum test measures differences between two samples by ranking the scores from each sample from high to low and then counting the number of ranks that each sample has above the other. This method is only appropriate if the difference between pairs is equal in all directions (i.e., an increase in one number will always cause an equal decrease in the other).

The signed rank test, on the other hand, allows researchers to assess the differences between the two groups even when the pairs are not all equal. That is, a small increase in one number may cause a larger decrease in the other. This test does not look at the absolute values of the differences but rather the sum of differences (signed) from the two paired sets.

Both versions of the test are based on the same null hypothesis of no difference between the two paired sets. If the calculated value of the test statistic is larger than the critical value, then the null hypothesis is rejected and there is evidence of a statistically-significant difference between the two sets.

The Wilcoxon test is a useful tool for researchers who want to make quick and reliable inferences about paired data. It is an easy-to-interpret statistic and is best used in situations where data samples are related in some way and the underlying distribution of the data cannot be assumed. It is a helpful tool when comparing pre-test and post-test scores, stock prices, and any other data that is collected in pairs.