Whether to use nonparametric tests to compare two groups when sample size is large but assumptions are violated I have a data set that has 600 observations divided in two groups.  I am going to compare the central tendencies (e.g., the means) of these two groups.  However, there are violations of classical assumptions present, such as normality and equality of variances.  


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*Can I use a parametric approach (specifically, the t-test), since the sample sizes are large (based on the Central Limit theorem), or do I have to use a non-parametric approach?  

*If I should use a non-parametric approach, which test (Mann-Whitney, Median or Kolmogorov-Smirnov) is most appropriate?

 A: With 600 observations divided between only two groups (assuming that they're divided fairly equally, not 598 and 2), you most likely have enough data to feel comfortable using the t-test.  If the variances differ between the two groups, you would want to use the Welch-Satterthwaite correction for the effective degrees of freedom.  
However, you might want to use the Mann-Whitney U-test anyway.  The U-test is more powerful than the t-test when the data are not normally distributed.  Best of luck with your project.  
A: I would like to add only one remark to gung's answer which is of course in general correct. 
Central Limit Theorem will sooner or later "kick-in" only for distributions that have finite moments. If your data was generated by a process described by a probability distribution with infinite moments (for example power distribution/pareto distribution), then CLT will never work (i.e. regardless of the size of your sample). So if you suspect that your data may not have finite moments, then I would say it is safer to use nonparametric tests (like Mann-Whitney's U test).
