Mann-Whitney Test

Mann-Whitney Test: n_a= n_b=

The logic and computational details of the Mann-Whitney test
are described in Subchapter 11a of Concepts and Applications.

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=n_a+n_b 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

U_A
= n_an_b +
n_a(n_a+1)
2
— T_A

where T_A = the observed sum of ranks for sample A, and

n_an_b +
n_a(n_a+1)
2
= the maximum possible value of T_A
With relatively small samples, the calculated value of U_A can be referred directly to the sampling distribution of U_A. For cases where n_a and n_b both fall between 5 and 21, inclusive, the lower and upper critical values of U_A are calculated by this page and placed in the designated table below.
As n_a and n_b increase, the sampling distribution of T_A becomes a reasonably close approximation of the unit normal distribution. If n_a and n_b 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 n_a and n_b are both equal to or greater than 10.

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 n_a and n_b 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 Data_T

Sample A	Sample B	Import data to data cells Clear A Clear 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


If you are calculating from raw data, be prepared to wait a moment while the raw data are converted into ranks. The time required for this conversion will increase as a function of the sizes of the samples and the number of tied ranks.

Mean Ranks for
Sample A	Sample B	U_A =	P₍₁₎	P₍₂₎
		z =

Note that mean ranks are provided only for descriptive
purposes. They are not part of the Mann-Whitney test.

Home

Click this link only if you did not arrive here via the VassarStats main page.

Disregard these cells. They are merely place holders.

	Level of Significance for a
	Directional Test
	.05	.025	.01
	Non-Directional Test
	--	.05	.02
lower limit
upper limit


	U_A	= n_an_b +	n_a(n_a+1) 2	— T_A


	n_an_b +	n_a(n_a+1) 2	= the maximum possible value of T_A