# Mann-Whitney U Statistic seems very large - is something wrong?

I compared two unequal sized samples, N1 = 230 and N2 = 120 using a Mann-Whitney U test. The U is shown to be large such as 2829.

The p value is non-significant but I want to know if such large values of U is ok? Or is there something wrong there? I am using PAST software.

With large sample sizes the statistic may be quite large.

A typical definition of the U statistic is the number of cross-sample pairs where an observation from the first sample exceeds an observation from the second sample.

[That definition would result in an expected value (if the null hypothesis were true) of $n_1n_2/2$.]

There are several different (but closely related) definitions of the statistic that are used in various packages, which may have different expected values under the null. For example, some make it equivalent to the Wilcoxon-rank sum statistic, by adding $n_1(n_1+1)/2$. Others consider both the first statistic I mentioned and the number of times a value from the second sample exceeds one from the first sample (i.e. swapping which one is considered "the first" and which "the second") and then takes the smaller of the two statistics. Others may subtract the expected value

You would have to consult the help for your package to see which exact definition is used.

Looking at the manual for PAST here, is says (p47):

For each value in sample 1, count the number of values in sample 2 that are smaller than it (ties count 0.5). The total of these counts is the test statistic U (sometimes called T). If the value of U is smaller when reversing the order of samples, this value is chosen instead (it can be shown that U$_1$+U$_2$=n$_1$n$_2$)

This is one of the possible statistics I mentioned earlier.

(I note that elsewhere on this page of the manual, it makes a number of incorrect statements about the Mann-Whitney test. Exercise a great degree of caution when reading this manual.)

This definition for the U statistic would make the expected statistic somewhat below 13800 (about 13083). If the p-value is larger than 0.05 then I think you should not be seeing a U value below 12039.

I can't see how you're getting a value as small as 2829 without getting a very small p-value.

So in fact something does seem wrong, but it's not that the statistic is too large -- if everything you've said is correct, your U statistic is much too small.

• Thanks Glen for the explanation. I repeated the test in SPSS 20 and got the same results. While comparing N1= 230 and N2= 110, the U is 10802 with p= ,311. Also, comparing N1=230 and N3=78, the U is 2828 with a p=,ooo. Similarly, N2=110 and N3=78, U = 1351 with a p =.007. I guess this makes sense then. Please let me know your view. Jan 5 '17 at 6:33