Statistically significant differences, when sample is not normal.

Question

I have two sample which doesn't comes from normally distributions. How to analyse whether there are statistically significant differences ?

Answer 1

In special situations, you can still do exact inference. However, if you're looking for general approaches, then you can try:

1. Nonparametric tests and confidence intervals (there are TONS of methods in this class that do not depend on normality)
2. Bootstrapping (parametric or nonparametric)
3. Likelihood-iterval-based
4. Bayesian inference

So, you have a lot of areas to look into. Normal-based stats is "classical" stats, and these have been shown to be broadly useful even when the underlying distribution is known to be very non-normal.