Mann-Whitney test for a large sample I'm not a statistician, so pardon me for being naive on this subject.
I'm trying to understand if there's any statistically significant difference in the medians of 2 groups. Here are some of the salient features of my groups:

*

*Each of the groups has Millions of observations

*Each of the groups is not normally distributed

*The observations are continuous

*One of the groups has almost 15x the observations in the other group

*The groups are mostly independent of each other

If the groups were normally distributed, I could have used the T-test to figure this out.
So this leads me to believe that a Mann-Whitney test would be more useful in this case. But because I have Millions of observations in both the groups, I'm not sure if the Mann-Whitney test results will hold true. In one of the Stack Overflow posts, I read that Mann-Whitney test does not work well with so many observations.
Should I just take much smaller random samples from my 2 groups and perform the Mann-Whitney test many times and then look at the results?
Or is there a better approach to doing this? Any help would be much appreciated.
 A: First, the MW test is not a test of medians (there are some additional assumptions that make it a test of medians, but those same assumptions would also make it a test of means).
The Central Limit Theorem tells us that we can use normal base inference (e.g. the t-test) when the population is not normal, but the sample size is large enough.  A sample size of millions is probably big enough that the t-test will give a good enough approximation.
If you still want to compare medians, or still don't trust the t-test then you could use a permutation test (the MW test is a special case of a permutation test) or create a bootstrap confidence interval, or just replace the data with their ranks and do a t-test on the ranks (the MW approximation).
The comment above about looking at the meaningful differences is probably more important than the p-value in this case (with millions of data points you can find extremely small differences to be statistically significant).  Confidence intervals on the differences will probably be much more meaningful than p-values at those sizes.
