Will using Mann Whitney U test, comparing the medians, give a valid inference to the populations being compared? I am basically comparing the length and/or weight of an animal categorised by the presence or absence of a specific allele, so I have two groups. The data for length and weight do not seem to be normally distributed, both from the Shapiro Wilks test and visually looking at histograms. This is the still the case when I have transformed the data log10. Therefore I chose to use the non parametric test: Mann Whitney U test. As the shapes of the histograms for both groups are somewhat similar, I tended towards comparing the medians of the data. 
Is this correct so far? or have I missed something really obvious that i could do to compare the means? 
My question is: will comparing the medians allow me to infer that one group has a significantly greater length/weight than the other? 
Thank you or your time and I appreciate any response!! 
 A: (Moving comments to an answer so this doesn't go unanswered.)
(i) The Mann-Whitney doesn't compare medians (at least not without some assumptions) 
(ii) Choosing which test to do on the basis of tests of assumptions tends to result in the following tests not having their nominal properties (it also answers the wrong question); 
(iii) It's not clear to me why "similar shapes" implies "compare medians", nor why you think you have a choice in what you're comparing if you already chose the test -- how would the subsequent decision to compare medians come into it if you're already doing Mann-Whitney?  
(iv) It may be possible to compare means -- there are several possibilities. 


*

*using GLMs might be reasonable, such as a gamma GLM. 

*If you assume identical distributions under the null and some kind of alternative that one variable is stochastically larger (including location shift or scale shift as special cases) you could either do a permutation test to test equality of means against ordered means, as long as the means exist, or  

*with the additional assumptions in (2.) the Mann-Whitney should also   test equality of means against ordered means (as long as the means exist).
