For reliable results your data has to conform to the assumption of the statistical test you want to apply. If your data is not normally distributed you can either use distribution free methods (non parametric test) or check the distribution your data and apply that particular test.

No.

1) The distribution model is a theoretical thing. Real data will have a frequency distribution (empirical distribution), what can only more or less well approximated by some theoretical model. The question is never if the data has some particular distribution, but possibly if some distribution model may serve as a sufficiently good approximation to characterize the empirical distribution.

2) Tests are based on assumptions, and some tests assume a particular distribution model. If you use tests, the assumptions should be resonable. Otherwise, the interpretation of the results won't be reasonable either.

There are tests assuming a normal distribution of the statistic (not of the data! - but usually normal distributed data ensures normal distributed statistics). But there are also tests assuming different distributions, like the binomial distribution, the hypergeometric distribution, or the chi-squared distribution. The so-called non-parametric tests don't assume a particular distribution model that could be written down as a formula (with paramenters specifying particular properties of the distribution), but they do make assumptions about distributions as well (e.h. homogeneity in higher distributional moments [variance, skew, kurtosis]).

Interestingly, when the assumptions of a test are obviousely not reasonable for the data, the hypothesis to be tested is also not very reasonable. Often, thinking clearly about the hypothesis that should be tested will propose a test which assumptions are ofetn very reasonable for the given data.

Mugumaarhahama Yannick, nope. Not all "parametric methods" (methods based on a parametric distribution model) "require normal distributed data". There are other parametric distribution models (Binomial, Poisson, Gamma (including Chi²), Beta, Weibull, Gompertz, ...). It's also a formulation that is prone to be misunderstood. The methods don't "require" such data. They are based on assumptions about such distributions. This assumption should be reasonable.

Skewness and kurtosis values more than normal distributions skewness and kurtosis value