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It seems there are two formulas to perform a wilcoxon signed rank test - one where the sum of all the signed ranks is w and another where the minimum of the sum of the negative ranks and positive ranks is used. Could someone please explain the difference and which one is the more appropriate method?

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Ranks range from 1 to N. The sum of all ranks is a constant N(N+1)/2. So there are mathematically equivalent ways to express the test. I am not sure that you have properly described the two equivalent formulations.

As an aside a different rank test the Wilcoxon-Mann-Whitney rank sum test has the formulation in terms of the sum of the ranks for the sample from say population A (there are two populations A and B that are being compared) and the not immediately obvious Mann-Whitney formulation in terms of probabilities of the from P{X$_i$ > Y$_i$) where X$_i$ comes from population A and Y$_i$ from population B. I think for a time people did not realize that these are equivalent tests.

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