Statistical Inference: A Short Course

Appendix 10.B Wilcoxon Signed Rank Test (of a Median)

Suppose we have evidence that points to a symmetric population distribution. Then a Wilcoxon signed rank test for the median is in order. Additionally, with the population distribution symmetric, this test for the median is “equivalent to a test for the mean.” To execute the Wilcoxon signed rank test, let us extract a random sample of size n with values X₁, X₂, . . ., X_n from a continuous symmetric population and test the following hypotheses pertaining to the population median MED:

Case 1	Case 2	Case 3
H0: MED = MEDo	H0: MED = MEDo	H0: MED = MEDo
H1: MED ≠ MEDo	H1: MED > MEDo	H1: MED < MEDo

where MED_o is the null value of MED.

To perform the Wilcoxon signed rank test, let us consider the following sequence of steps:

1. Subtract the null value MED_o from each X_i to obtain Y_i = X_i − MED_o, i = 1,. . .,n. (If any Y_i = 0, then eliminate X_i and reduce n accordingly.)

2. Rank the Y_i's in order of increasing absolute value. (If any of the nonzero Y_i's are tied in value, then these tied Y_i's are given the average rank.)

3. Restore to the rank values 1, . . ., n the algebraic sign of the associated difference Y_i. Then the ranks with the appropriate signs attached are called the signed ranks R_i, i = 1, . . .,n, where denotes the rank carrying a positive sign.

Let us specify our test statistic as

(10.B.1)

the sum of the ...