Are differences normally distributed?
The set of differences between sample means is normally distributed. This will be true if each population is normal or if the sample sizes are large. (Based on the central limit theorem, sample sizes of 40 would probably be large enough).
What tests are not normally distributed?
A non parametric test is one that doesn’t assume the data fits a specific distribution type. Non parametric tests include the Wilcoxon signed rank test, the Mann-Whitney U Test and the Kruskal-Wallis test.
Is the calculated mean wrong for non-normally distributed data?
However if the samples are sufficiently large the Central Limit Theorem which guarantees approximate normal distribution for the mean can be applied for the adoption of parametric methods. The calculated mean and the standard deviation are not wrong for non-normal distributed data, nor do they lead to wrong results, as you wrote.
What can you do with normal distribution data?
Normal distribution is a means to an end, not the end itself. Normally distributed data is needed to use a number of statistical tools, such as individuals control charts, Cp / Cpk analysis, t -tests and the analysis of variance (ANOVA).
Do you need to be fixed with the normal distribution?
We don’t need to be fixed with the normal distribution. Stats software allows to fit models for a wide variety of probability models effortlessly. You can also try to normilize your data, if you have enough data it will be possible, then use parametric methods, which may work properly even with some non-normal data.
When does the distribution of data become an issue?
If a practitioner is not using such a specific tool, however, it is not important whether data is distributed normally. The distribution becomes an issue only when practitioners reach a point in a project where they want to use a statistical tool that requires normally distributed data and they do not have it.