Mann-Whitney Test

 The logic and computational details of the Mann-Whitney test are described in Subchapter 11a of Concepts and Applications.
For na= nb=
In order to apply the Mann-Whitney test, the raw data from samples A and B must first be combined into a set of N=na+nb elements, which are then ranked from lowest to highest, including tied rank values where appropriate. These rankings are then re-sorted into the two separate samples.

If your data have already been ranked, these ranks can be entered directly into the cells headed by the label «Ranks». In this case, please note that the sum of all ranks for samples A and B combined must be equal to [N(N+1)]/2. If this equality is not satisfied, you will receive a message asking you to examine your data entry for errors.

If your data have not yet been rank-ordered in this fashion, they can be entered into the cells labeled «Raw Data» and the ranking will be performed automatically. There is also an option, below, for importing raw data from a spreadsheet.

After data have been entered, click one or the other of the «Calculate» buttons according to whether you are starting out with ranks or raw data.

• The value of U reported in this analysis is the one based on sample A, calculated as
 UA = nanb + na(na+1)2 — TA

where TA = the observed sum of ranks for sample A, and
 nanb + na(na+1)2 = the maximum possible value of TA

• With relatively small samples, the calculated value of UA can be referred directly to the sampling distribution of UA. For cases where na and nb both fall between 5 and 21, inclusive, the lower and upper limits of the critical intervals of UA are calculated by this page and placed in the designated table below. If either of the samples is of a size smaller than 5, additional instructions will be given below.

• As na and nb increase, the sampling distribution of TA becomes a reasonably close approximation of the unit normal distribution. If na and nb are both equal to or greater than 5, this page will also calculate the value of z, along with the corresponding one-tailed and two-tailed probabilities. Note, however, that the approximation to the normal distribution is best when na and nb are both equal to or greater than 10.

Option for Importing Raw Data via Copy & Paste:T

Import Raw DataT
 Sample A Sample B Import datato data cellsClear A Clear B

Data Entry:T
 Ranks for Raw Data for count Sample A Sample B Sample A Sample B
 Mean Ranks for Sample A Sample B UA = P(1) P(2) z =
Note that mean ranks are provided only for descriptive purposes. They
are not part of the Mann-Whitney test. ~ Note also that the z-ratio is
calculated only if na and nb are both equal to or greater than 5.

Critical Intervals of UA for Q
 Level of Significance for a Directional Test .05 .025 .01 Non-Directional Test -- .05 .02 lower limit upper limit
The observed value of UA is significant at or beyond the designated level if it is equal to or smaller than the indicated lower limit for that level or equal to or greater than the upper limit. It is non-significant if it is larger than the lower limit and smaller than the upper limit.
The adjacent critical intervals are calculated only if na and nb both fall between 5 and 21, inclusive. For sample sizes smaller than 5, you can refer your results to a standard table of Mann-Whitney critical values, such as the following, provided by the Department of Mathematics & Statistics at the University of Saskatchewan:
with na=, nb=, UA=, and UB=

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