Uses of the Mann-Whitney U Test for Business

The Mann-Whitney U test is a nonparametric test of the equality of medians of two populations.  Although the test does not require normal populations, it does require equality of variance.  In its pure nonparametric form, the test requires a table of U values with which to compare two obtained U statistics from two independent populations (Pagano 232-237).  Since your textbook does not have this table, we have to take a different approach.

Some authors have adapted this test to the z baseline by using a z conversion formula.  In its modified form, the test is called the Mann-Whitney test.  It is used in business for evaluating satisfaction ratings on goods and services.  In fact, for any study about two independent populations, unknown shapes of the parent populations, samples of size 11 or more from those populations, and equal variances of the populations from which the samples are selected, we may use the Mann-Whitney test.   Below is an example of this test.

Example: “Will the Better Show Step Forward”

A television producer must decide which of two shows to sponsor for next season’s TV lineup.  To make his decision, he invites 24 randomly selected people to a preview of the shows.  Half the participants see show 1, and the other half see show 2.  We’ll assume that the shows have titles for identification not numbers.  Participants are told to rate each show on such characteristics as story line, overall acting, acting of each performer, and so forth.  Also, we’ll assume that the rating scale ranges from 0 to 60 and that the ratings are independently done.  In other words, participants are not collaborating.

The producer hires a statistician to conduct the research and process the data.  The statistician decides on the Mann-Whitney test for decision-making.  The procedure below is inspired by Doane and Seward (706-707).  Notice the stages that the raw data goes through.  Also, witness that the two samples represent the two populations of ratings for Show 1 and ratings for Show 2.

Participant	Rating	Show	Rank	Participant	Rating	Show	Rank	
1	28	1	1	13	46	2	13	
2	30	1	2	14	47	1	14	
3	32	2	3.5	15	48	1	15	
4	32	2	3.5	16	50	2	16.5	
5	38	1	5	17	50	1	16.5	
6	40	1	7	18	51	1	18.5	
7	40	2	7	19	51	2	18.5	
8	40	1	7	20	52	2	20.5	
9	41	2	9	21	52	2	20.5	
10	42	1	10	22	56	2	22	
11	45	1	11.5	23	58	2	23	
12	45	1	11.5	24	60	2	24	

Ties are calculated as the average of the positions.  For example, two participants were tied on their ratings for the third and fourth places.  So, = 3.5.  Another example is that the rating 40 shows up three times.  Therefore, the allocated positions are 6, 7, and 8.  What’s the average of these three numbers?  Why it’s 7.  Once a position on the rating scale is used up, the rank goes to the next place in the sequence.

The next step is to split apart the shows by their Satisfaction and Rank.

Show	1	Show 	2	
Satisfaction	Rank	Satisfaction	Rank	
28	1	32	3.5	
30	2	32	3.5	
38	5	40	7	
40	7	41	9	
40	7	46	13	
42	10	50	16.5	
45	11.5	51	18.5	
45	11.5	52	20.5	
47	14	52	20.5	
48	15	56	22	
50	16.5	58	23	
51	18.5	60	24	

Before the Mann-Whitney test is performed, the researcher should check the assumption of equal variances.  The test is called the F test for Homogeneity of Variance.  It is a special F test that splits alpha for the two-tailed type.  (This test is different from the omnibus F test conducted in one-way ANOVA and the F tests for factorial ANOVA.)  Nonetheless the test does compare two variances with the larger variance always assigned to the numerator.  Degrees of freedom for both groups are each sample size decreased by 1.  The obtained F ratio must exceed a critical F for significance.  This is one of the tests where researchers hope for non-significance so that they could proceed with the main test.

H0:   (Population variances are the same.)

H1: 

The research hypothesis says that the samples suggest both population variances are different.

Since it’s the variances that we want to compare, the easier approach is to apply Excel’s  VAR function than to use its Descriptive Statistics program.  See the results of the Excel run on page 3.

Test Statistic: F =  = 1.664965845 = 1.6650 rounded to ten-thousandths.

Ratings for Show 1	Ratings for Show 2	
28	32	
30	32	
38	40	
40	41	
40	46	
42	50	
45	51	
45	52	
47	52	
48	56	
50	58	
51	60	
 Variance for 1	Variance for 2	
53.45454545	89	

Critical  is between 3.4296 and 3.5257.  Test is at alpha = 0.05 and is split evenly in both tails of this asymmetric curve.  Critical value is from Triola (777).  Your textbook is not set up for this kind of test.  Rather, the F table accommodates only ANOVA tests.  However, we often do not need to make comparisons where F tests are concerned, since values around 1 are almost always non-significant, while values much higher than 1 are usually “sure shots” for significance.  For assignments or tests, we’ll waive your check of this assumption.

Since the null hypothesis is supported, we could move on to the problem at hand.

Calculations:  Rank Sum 1 = 119	Rank Sum 2 = 181

		 = Sample Size 1 = 12	 = Sample Size 2 = 12
		
		 = Mean Rank 1 = 9.917	 = Mean Rank 2 = 15.083

H0:  (There is no difference in satisfaction)

H1:  (There is a difference in satisfaction.)

Test Statistic: z = = -1.790 rounded

Note that 12 in the general formula is a “carved in stone” part of the z conversion formula itself.  It is a coincidence that the sample sizes are also 12.

With a two-tailed z-test at the 5% level, we have to compare the obtained z to 1.96, the critical values.  Since z = -1.79 is not more extreme than -1.96, the null hypothesis is the more attractive explanation.

Summarizing, the first test was a preliminary test of an assumption for the model.  The outcome of this test, namely to support the null hypothesis of equal variances, gave the researcher the green light to go ahead with analysis by the Mann-Whitney test. The main test, then, suggested that neither show is a clear-cut favorite. 

Consequently, this outcome tells the television producer that a population of people similar to those in the sample would not favor one show over the other.  Were this the outcome of a real-life statistics test, the producer would be left wondering which show to select for televising.  His choices would be to make a decision by himself or to take a new, larger sample.  Of the two choices, the second is wiser.  When sample sizes are larger and more diversified, we typically should get winner, since larger samples better represent the population.

Works Cited

Doane, David, and Lori Seward. Applied Statistics in Business and Economics.
	New York: McGraw-Hill, 2007.

Pagano, Robert. Understanding Statistics in the Behavioral Sciences, 3rd ed.
	St. Paul, MN: West,1990.

Triola, Mario. Elementary Statistics. Boston: Pearson/Addison-Wesley, 2006.