Since many statistical procedures (e.g. t-test, ANOVA, Pearson’s r (for efficient estimates)) require the normal distribution of the tested variables ('normality-assumption') one may ask if (at least) some version of the central limit theorem is applicable to all procedures which require this assumption. Personally, I'm not sure, what's the right answer. This uncertainty can be traced back to two factors:
First, statistical textbooks provide different and partly inconsistent information about this topic. For example, in nearly every source I read CLT is said to apply for t-tests with sufficient large samples in such a way that the 'normality-assumption' is fulfilled or doesn't have to be tested. Some of those textbooks don't claim this for ANOVA or Pearson's r and insist on testing this assumption regardless of sample size. In contrast other textbooks/sources (e.g. persons posting in CrossValidated) claim the applicability of CLT in those procedures (resulting in the same consequences for testing the requirements like in t-tests).
Second, while the applicability of CLT is clear to me for (simple) sums of random variables and consequently the mean of such variables, the different and more complex computations of the test statistics (t, F, r) of the mentioned procedures simply raise the question, if these statistics can be understood as a manifestation of the CLT for sums of random variables.
To sum up, my questions are:
1: Does (at least some sort of) CLT apply to the 'normality-assumption' of all mentioned statistical procedures (or even not only to the mentioned)?
2: What are the reasons and backgrounds for the answer to 1:?
3: Can you recommend any statistical source, which examines requirements of (common) methods of applied statistics comprehensively? My experience (by now with many sources) is that some requirements are either missing or not explained in a didactically coherent and usable manner.
Thanks a lot and best regards, Stefan
