**Mann-Whitney Non Parametric U Test**

The preferred non-parametric method for unpaired samples is the Mann-Whitney non parametric hypothesis test or Mann-Whitney test (it is also called as Wilcoxon Rank Sum Test or the Mann Whitney Wilcoxon Test) and thus the non parametric solution to evaluating two independent datasets comparable to the Student’s T test.

Mann-Whitney test is a non-parametric test that is to compare two sample means that may come from the same population, and used to test whether two sample means are equal or not. This is a powerful non parametric test, and is an alternative to the t- test when the normality of the population is either unknown or believed to be non normal.

Few researchers also interpret Mann-Whitney test will helps to compare the medians of the two populations. Usually parametric tests will compare the means (Null hypothesis: μ1=μ2) between the independent groups. Whereas, the null hypotheses and the two-sided hypotheses non-parametric tests can be indicated as follows:

- Null Hypotheses H
_{0}: Two populations are equal - Alternative Hypotheses H
_{1}: Two populations are not equal.

The Mann-Whitney test is regularly performed as a two-sided test, therefore the investigate hypothesis indicates that the two populations are not equal, instead of specifying the directionality.

**Assumptions of the Mann-Whitney**:

- The sample drawn from the population is random.
- Independence within the samples and also mutual independence is assumed. That means that an observation is in one group or in the other (it cannot be in both).
- Identical (non-normal) distributions
- Ordinal measurement scale is assumed.

**Uses of Mann-Whitney**

Mann-Whitney U test is can be used for every industry. But is more frequently used in healthcare, nursing, business, and also many other disciplines. In medicine, it is more efficient method to know the effect of two medicines and whether they are equal or not. It is also used to know whether or not a particular medicine cures the ailment or not.

**Procedure to conduct Mann-Whitney test**

- Combine two samples data together, sort data in ascending order
- Convert data to ranks (1, 2, 3,… Y)
- Separate ranks back in to two samples
- Compute the test statistic, U
- Determine critical value of U from Mann-Whitney table
- Finally, Formulate decision and conclusion

**Calculation of Mann-Whitney Test**

**Case 1: **For comparing two small sets of observations (samples sizes less than 20)

U_{1} = n_{1}n_{2}+0.5n_{1}(n_{1}+1)-R_{1}

U_{2 }= n_{1}n_{2}+0.5n_{2}(n_{2}+1)-R_{2}

Where
U_{1}+U_{2}=n_{1}n_{2}

- n
_{1}is the observations from first population - n
_{2}is the observations from second population - R
_{1}is the sum of observation ranks for first population - R
_{2}is the sum of observation ranks for second population

Finally, calculate the U statistic as the smaller of U_{1} and U_{2}. U= min (U_{1,} U_{2})

**Case2: **Normal approximation and tie correction

For sample sizes above ~20, the distribution of U rapidly approaches to the normal distribution

U mean = μ_{U} =0.5 n_{1}n_{2}

## Example of Mann-Whitney Test

A researcher while conducting studies on Biomass of various trees. He wished to determine if there was a difference in the biomass of male and female Juniper trees. So, he randomly selected 6 individuals of each gender from the field. He dry them to constant moisture, chip them, and then weigh them to the nearest kg.

- H
_{0}: There is no difference between biomass of male and female Juniper trees– Biomass_{male}= Biomass_{female}(medians are equal) - H
_{1}: There is a difference between biomass of male and female Juniper trees– Biomass_{male}≠ Biomass_{female}(medians are not equal)

The data are ratings (ordinal data), and hence a non-parametric test is appropriate – the Mann-Whitney U test (the non-parametric counterpart of an independent measures t-test).

- n
_{1}=6 - n
_{2}=6 - R
_{1}= 1.5+3+5.5+7.5+4+1.5=23 - R
_{2}= 5.5+11+10+12+9+7.5=55

And then calculate U_{1} and U_{2}

- U
_{1}= n_{1}n_{2}+0.5n_{1}(n_{1}+1)-R_{1}= 6*6+0.5*6(7)-23=34 - U
_{2 }= n_{1}n_{2}+0.5n_{2}(n_{2}+1)-R_{2}= 6*6+0.5*6(7)-55=2

U=
min (U_{1,} U_{2}) = 2

**Mann-Whitney table: For two-tailed test 5% significance level **

U_{critical} = 5

Calculated U is value less than the critical value of U for a 0.05 significance level. U_{calculated }< U_{critical} . Hence, we can reject the null hypothesis.

So, we can say there is a highly significant difference between male and female Juniper trees’ biomass.

**Six Sigma Black Belt Certification Mann-Whitney Test Questions: **

**Question:** Which of the following test, calculates by rank all the participants scores from lowest to highest and adding up the ranks separately for each of the group.

(A) Wilcoxon signed rank test

(B) Kruskal Wallis test

(C) Mann-Whitney test

(D) Friedman Test

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## Comments (8)

“Mann-Whitney test is a non-parametric test that is to compare two sample means”

How come Non-parametric test and used to compare means?

Hi Mohamed,

I think you’re asking why you would use a non-parametric test to test means. I’ll answer that. If you had a different question in mind, just leave another comment.

You would use a non-parametric test of means when the distribution you’re testing against isn’t normal.

Best, Ted.

“Calculated U is value less than the critical value of U for a 0.05 significance level. Ucalculated Shouldn’t it means reject null instead?

I agree with you. The following sentence means that the null hypothesis was rejected.

> So, we can say that seem like there is a highly significant difference between biomass of male and female Juniper trees.

Thank you.

Those disagreements between the conclusions in the same articles are really annoying. Led to massive misunderstanding!

Thank you, we have corrected the statistical conclusion.

Hi Ted

I have juste a question please

I observe that there is a lot of reference of certification

Like IASSC, ASQ, 6 sigma study , councel six sigma

in your point of views which certifications better ,

and what reference should we avoid

Youssef, It really depends on the individual. I have a comprehensive article here.