Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric test used to compare the distribution median with a given value m. This criterion could be used as an alternative for one-sample Student t-test. Unlike the t-test, Wilcoxon signed-rank test can work with non-normal distributions.

This test has the following requirements:

the scale of measurement^[1] should be ordinal, interval or ratio (i.e. test could not be applied to nominal variables)
the distribution should be continuous and symmetric relative to its median
number of distinct values in the X array should be greater than 4

Note #1
The distribution symmetry requirement is critical. If the distribution is non-symmetric, the test could not be used. In that case we can use less powerful (but more general) sign test.

Subroutine WilcoxonSignedRankTest returns three p-values:

p-value for two-tailed test (null hypothesis - the median is equal to the given value)
p-value for left-tailed test (null hypothesis - the median is greater than or equal to the given value)
p-value for right-tailed test (null hypothesis - the median is less than or equal to the given value)

The test algorithm is simple. All elements which are equal to m are thrown out. After that we have elements of two types only: elements which are greater than m and which are less than m. Elements are sorted by their absolute value. After that W⁺ (sum of positive elements ranks) is calculated. If this hypothesis is true (the median equals m), W⁺ has a distribution which could easily be calculated (using dynamic programming) and tabulated. To define the significance level corresponding to the W⁺ value tables (for small Ns) or asymptotic approximations (for greater Ns) are used. This method lets us calculate p-values with two decimal places in interval [0.0001, 1]. "Two decimal places" does not sound very impressive, but in practice the relative error of less than 1% is enough to make a decision.

Note #2
Some sources recommend to use normal distribution with μ = 0.25·N·(N+1) and σ² = N·(N+1)·(2N+1)/24 when estimating the significance level. In fact, as N increases W⁺ converges to normal distribution with these parameters. However, although the rate of convergence is good it's better not to approximate W⁺ with normal distribution, because this approximation can be insufficiently precise.