What method / technique I should use to compare two groups of people when the data is non-normally distributed data? I am going to compare means of two groups (Male, Female) from my complex stratified sample.
I have scores of 3 subjects of more then 3000 respondent for each subject. 
I checked for normality and get the following picture:

i.e data for groups isn't normal in all three cases. 
Therefore I can not use the tests of hypothesis of homogeneity  which assume normality, also randomness of sample.
Nevertheless I want to compare who is better boys or girls? How can I do this correctly and without bias? 
Edit: By the way there are no intersections in the three set of respondents. I visualize the three because there is the one problem for all three subjects.
May be I ask the silly question but I guess when normality and randomness conditions are violated, it is incorrect to  use such methods. Am I not right?
 A: Visualization would help, often the eye can see and understand more through graphics than the numbers. I can picture what your outcomes look like, but understanding what the score comes from would help. 
The min/max and mean match appropriately. The small adjustment made to the median let us know the sample is skewed, but since the adjustments are small, the skew isn't too influential (though it has shown to be significant by some rule of thumb ideas). The Kurtosis is the biggest problem it seems like. I picture a bell that has a peeked top possibly with a divit in the middle, the drops rapidly and goes into two low population tails, with one of them skewed. This suggests that an overwhelming majority of people test within a small range, and the other groups differ significantly. It may help to identify what differentiates the small groups the are significantly above and below the mean. It may make sense to take them out of the sample, with the explanation that you want to assess differences in individuals, and to avoid the undue influence of X and Y you have constrained your sample to not include them. I'm guessing you would then have a sample with a smaller range, but a more normal distribution. Without more information on the measures, it is difficult to say more than that. Look up tests that are robust to violations of the assumption of a normal distribution, such as generalized estimating equations. 
