system
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human
What does data normality imply?

I see a lot of statistical papers and tutorials checking for data normality before doing any statistical test. I know that some tests assume that the data is normally distributed, but I've seen people doing this even when doing a linear regression. So my questions are:

1. What data normality imply in an statistical analysis?
2. Why statistical tests assume normality when it doesn't happen so often in the reality?
3. When to check for data normality?

Thank you, guys! This subreddit is awesome!
gpt
>  I've seen people doing this even when doing a linear regression

The usual tests, CIs and prediction intervals in linear regression also assume normality -- but not of any of the observed variables (rather, of the unobservable errors).

> What data normality imply in an statistical analysis?

The data themselves are not actually normal; normality is an assumption about a population, which is an assumption of some procedures.

> Why statistical tests assume normality when it doesn't happen so often in the reality?

It's rather a good question. You can often find people bringing up something about the central limit theorem but that misses a couple of important issues.

Nonetheless, some of the tests that assume normality are at least somewhat level-robust to moderate deviations from it (that is, the significance level is not badly impacted in some circumstances), though power can be an issue sometimes.

> When to check for data normality?

Tests answer the wrong question. 

However, if you're choosing your analysis based on looking at the data (whether you use tests or not), you're screwing with the properties of the analyses you're choosing between. So if you're going to use some sort of check on nearness to normality don't do it on the data you want to use some other test on.