# Mann Whitney Test

#### nealprakash

##### New Member
I have been reading many statistical websites stating that the Mann Whitney Test is a test of medians. However, I believe that this is not really true? It is a test of the difference in the ranks. The Mann-Whitney only tests for a difference in medians when you assume that the only difference in the distributions of the two samples is the location, and not the scale or shape, of the distribution, which is very often too strong an assumption. Additionally, if one does make this assumption, then I believe it's also fair to say that the Mann Whitney test compare the difference in means as well.

I have a few questions that relate to this:
1. I am a bit confused why, often in research papers, the medians are reported when stating the results of a Mann Whitney Test. It seems that it's only under rare occasions when the Mann Whitney Test is actually able to compare the medians. Additionally, do researchers even check to see whether distributions are equal in the first place before saying that the test is a comparison of medians? If not, then it seems a bit unsound to report the medians.
2. Is the Mann Whitney test comparing the distribution of the ranks for two groups?
3. I am also a bit confused about what is stated here. The first post states that "the location-difference measure that the Mann-Whitney 'sees' is neither difference in means nor difference in medians -- it's the median of cross-group pairwise differences (the between samples quantity is the relevant estimate of the corresponding measure between populations)". Reporting Results of Mann-Whitney U Test - Means vs Medians . How exactly does "the median of cross-group pairwise differences" relate to ranks.

#### obh

##### Active Member
I have been reading many statistical websites stating that the Mann Whitney Test is a test of medians. However, I believe that this is not really true? It is a test of the difference in the ranks. The Mann-Whitney only tests for a difference in medians when you assume that the only difference in the distributions of the two samples is the location, and not the scale or shape, of the distribution, which is very often too strong an assumption. Additionally, if one does make this assumption, then I believe it's also fair to say that the Mann Whitney test compare the difference in means as well.

I have a few questions that relate to this:
1. I am a bit confused why, often in research papers, the medians are reported when stating the results of a Mann Whitney Test. It seems that it's only under rare occasions when the Mann Whitney Test is actually able to compare the medians. Additionally, do researchers even check to see whether distributions are equal in the first place before saying that the test is a comparison of medians? If not, then it seems a bit unsound to report the medians.
2. Is the Mann Whitney test comparing the distribution of the ranks for two groups?
3. I am also a bit confused about what is stated here. The first post states that "the location-difference measure that the Mann-Whitney 'sees' is neither difference in means nor difference in medians -- it's the median of cross-group pairwise differences (the between samples quantity is the relevant estimate of the corresponding measure between populations)". Reporting Results of Mann-Whitney U Test - Means vs Medians . How exactly does "the median of cross-group pairwise differences" relate to ranks.
Hi Neal,

1. When both distributions have a similar shape you may also say it compares the medians.
When both distributions are symmetrical, the median and the mean are similar.
2. Correct, the Mann Whitney U test checks the difference in the ranks.
3. I assume this says something like 2., as when you rank you compare every pair of observations ...

#### nealprakash

##### New Member
Hi Neal,

1. When both distributions have a similar shape you may also say it compares the medians.
When both distributions are symmetrical, the median and the mean are similar.
2. Correct, the Mann Whitney U test checks the difference in the ranks.
3. I assume this says something like 2., as when you rank you compare every pair of observations ...
Thank you. I have just one more question. For the situation, where I would want to extend the interpretation of the Mann Whitney test to compare medians (when the distributions for groups are equal, how would one go by checking to see if the distributions are equal? I can think of three ways, a QQ plot, a visual inspection of the two distributions, or a KS test. However, I am not sure what the convention is.