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.

Option for Importing Raw Data via Copy & Paste:T
Within the spreadsheet application or other source of your data, select and copy the column of data for sample A. Then return to your web browser, click the cursor into the text area for sample A and perform the 'Paste' operation from the 'Edit' menu. Perform the same procedure for sample B. In importing raw data into the Mann-Whitney test, it is absolutely essential that the numbers of data items for samples A and B are precisely the same as the values of na and nb that you have specified in setting up this page. For each sample, make sure that the final entry in the list is not followed by a carriage return. To perform this check, click the cursor immediately to the right of the final entry in the list, then press the down-arrow key. If an extra line is present, the cursor will move downward. Extra lines can be removed by pressing the down-arrow key until the cursor no longer moves, and then pressing the 'Backspace' key (on a Mac platform, 'delete') until the cursor stands immediately to the right of the final entry.T

Import Raw DataT
Sample A Sample B Import data
to data cells

Clear 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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©Richard Lowry 2001-
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Disregard these cells. They are merely place holders.