I am a high school Psychology student. I just performed a study where two separate groups of sizes 26 and 28 took the same test under different conditions. For the data analysis my professor has instructed me to use the Mann-Whitney test, which is unfortunate because both have sample sizes greater than 20. I saw something on the Wikipedia article about how the sample distributions are approximately normal at that point, but I have no idea how to use the z variable it gives me because I hardly know anything about statistics. Can somebody give a noob a simple process for performing the Mann-Whitney test correcting for the sample size?
Solved – Mann-Whitney (U) Test at N>20
large datapsychologywilcoxon-mann-whitney-test
Best Answer
The unpaired t-test is efectively "how to use the z variable" provided by the normal approximation, as you asked. In many circumstances the t-test (based on underlying normal distributions) may be as good as or better than the Mann-Whitney test, and as @AdamO notes the two tests tend to give similar results in practice.
But you have been asked to use the Mann-Whitney test, and there's no reason not to give it a try. There's no theoretical problem in using the Mann-Whitney test with samples of this (or any) size, and there's no need to "correct" for sample size. For example, the accepted answer on this page argues "you can't shoot yourself in the foot by using the Mann-Whitney test instead of the t-test, but the converse is not true." The Mann-Whitney test is based on comparisons of ranks among cases, rather than the actual values. This means it doesn't make strong assumptions about the underlying distributions, but it does raise some interesting issues having to do with ties, as discussed on this page. Modern statistical packages have ways of taking ties into account.
As you are just starting to learn how to apply statistics to real problems, you might find it informative to use both the Mann-Whitney test and an unpaired t-test on your data. Also, you should know that most analysts wouldn't consider this a "large" data set; that usually means more like thousands of cases (some might say millions) and up. But yes, this is the size of samples for which normal approximations are often good enough.