The distribution of the mean of a sample extracted from every type of distribution is always normal

Noteworthy

1. The sampling distribution get’s narrower as the sample increases
2. This is the ultime reasons why measuring more samples leads to smaller p-values

• Mean of the sampling distribution: \(\mu_{pop}\)
  
• Variance of the sampling distribution: \(\frac{\sigma}{\sqrt(N)}\)

Where \(N\) is the size of the sample

We are often dealing with data non normally distributed

1. Count based technologies never return negative numbers … (MS!)
2. Presence of sub-populations
3. Outliers

Sub populations are typically present in complex experiments

1. Non-parametric approaches (Kruskall-Wallis, quantile-regression, permutations, bootstrap, …)
2. Variable transformations

Remember that non parametric tests suffer of lack of power and are often completely useless fin investigations with only few samples (5-10)

Mean and median are ok!

1. Normality tests … again statistics ;-)
2. Graphical methods q-q plots

Quantile quantile plots are used to visually compare the theoretical quantiles of a distribution with the sample quantiles

Before running your statistical machinery give a look to your data to check if what you are doing makes sense …