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Wilcoxon Signed Rank Test

The Wilcoxon Signed Rank Test is a nonparametric equivalent of the one sample t-test. It tests the hypothesis:

H0 the mean equals zero
H1 the mean [is less than/greater than/not equal to] zero

The test involves:

• sort the values into order of their absolute magnitude (ignoring the signs) and allocate ranks
• calculate the sum the ranks of the data values are positive (S+)

An example would be:

 Rank 1 2 3 4 5 6 Data Value -1 -2 2.5 3 3 3.5

If the mean is non-zero the positive values will be clustered at one end or other of the ordered data set and S+ and S- will be very different.

For the example S+ = 3 + 4 + 5 + 6 = 18

For small data sets values the critical value is found from published tables. Because the sum of the ranks must be an integer, it is not usually possible to find the exact critical value:

 where

For larger data sets the z-statistic can be found from:

This gives the upper tail test (the mean is less than zero). For the lower tail test, use the sum S+ and for the two tailed test use both, remembering to use the table value twice.

The p-value can be calculated using normal distribution tables.

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