Tool:       The signed rank test is a nonparametric test of the null hypothesis that the unknown mean, μ, of a process is equal to the known mean, μ0, of another process:

        H0: μ = μ0

        H1: μ ≠ μ0

To use the signed rank test, we require a symmetric probability density function, (i.e., distribution), for observations so the median and the mean are equal.

First, we sample the new process N times, obtaining observations xi. From the xi and the known mean μ0 of the second process, we compute the distances, di, of the observations from μ0:

Second, we discard samples equal to μ0, if any, and reduce N accordingly.

Third, we rank the distances, di, in order of their absolute value (i.e., magnitude) from lowest, (rank = 1), to highest, (rank = N). If two or more di have the same magnitude, they are all assigned the average value of their ranks. If four di's have the same magnitude and correspond to ranks 3, 4, 5, and 6, for example, then all four di's are assigned a rank of (3 + 4 + 5 + 6)/4 = 4.5.

Fourth, we compute the sum, w−, of the ranks for negative di, and the sum, w+, of the ranks for positive di.

        w−   ≡   sum of ranks of di < 0

        w+  ≡   sum of ranks of di > 0

Fifth, we define w to be the smaller of w− and w+:

        w    ≡   lesser of w− and w+

Sixth, we compare w with a critical value, wa, from a table, such as Table A.17 of [1], for the desired significance level, α. If w ≤ wa, then we reject H0.

Note:       We can also test one-sided alternative hypotheses as follows:

H1: μ < μ0         reject H0 if w+ ≤ wa

H1: μ > μ0         reject H0 if w− ≤ wa

Definitions:

        μ0   ≡   known mean of existing process

        μ     ≡   unknown mean of new process

        N    ≡   total number of observations (data points) made

        i      ≡   index designating which sample (treatment) is being considered

        xi    ≡   value measured for observation (data point) i

        di    ≡   distance of observation (data point) i from known mean, μ0

        w−   ≡   sum of ranks of observations with negative distances

        w+  ≡   sum of ranks of observations with positive distances

        w    ≡   lesser of w− and w+

        wa  ≡   critical value of w for rejecting H0

        α    ≡   significance level for rejecting null hypothesis

Deriv:      Following the description presented in [2], the signs of the di may be thought of as outcomes of Bernoulli trials. That is, the signs are either positive or negative. If μ = μ0, then positive and negative signs are equally likely. Thus, p = q = 1/2 for the attendant Binomial distribution. (Note that this requires a symmetric distribution so that the mean and median have the same value.)

Taking the logic a step further, the xi are equally likely to have any rank. This means that the probability of any one particular pattern of + and − signs for the ranked distances is 1/2N.

If we start with the most extreme patterns in terms of w− or w+ values, such as all −'s or all +'s and work toward the least extreme pattern of half −'s and half +'s, we can determine what value of w− or w+ encompasses a set of patterns with total probability as close as possible to α without exceeding it. This yields the critical value, wa.

Note:       Finding wa is straightforward but tedious. Thus, it is convenient to look up wa values in a table.

Ref:    [1] Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying Ye, Probability and Statistics for Engineers and Scientists, 8th Ed., Upper Saddle River, NJ: Prentice Hall, 2007.

            [2] Anthony J. Hayter, Probability and Statistics for Engineers and Scientists, 2th Ed., Pacific Grove, CA: Duxbury, 2002.