Descriptive statistics

Rationale: When data are not normally distributed and the sample size is small (n < 30), we cannot use the central limit theorem and use Z or T test statistics. We rely on Non-parametric methods. There are two non-parametric tests, Sign Test and Wilcoxon Signed Rank Test, used as alternatives to Z and T test statistics.

Example: A given drug available on the market is known for long time to calm headache in 30 minutes. A researcher in a given drug company develops a new drug which could calm headache more rapidly than the previous one on the market. This new drug is administered to 8 patients and results are given below.

Question: Is there enough evidence to conclude at 5% significance level that headache disappears in less than 30 minutes?

Response: Define the difference variable D = X - 30, that is:

Wilcoxon Signed Rank Test

The sign test is simple but it has one major weakness. It does not take into account the magnitude of each difference.

Arrange the differences in order of absolute value (ascent ordering). Ignore d = 0 (n is reduced). If a group of observation has the same value, then compute the range of ranks and assign the average rank for each observation in the group.

Sign Test

Wilcoxon Signed Rank Test

signrank Time = 30