How is CLT related to the condition of data (normality assumption)? I apply statistical methods in sociological researches and now I feel a bit confused as I found out more about CLT.
For instance, if I have sample of 1000 observations, do I even need to check it for normality as CLT states that if sample is big enough, our data approximates normal distribution? And if it holds up, why people constantly plot q-qplot or do tests like shapiro to test for normality? (Plus we know that many of stat methods such as ANOVA, t test quite robust to violation of normality)
Of course, when population is normally distributed and we have big enough sample we can be sure that our sample is normally distributed, but what if it is not, but I still have big sample(e.g. 1000), should I really worry about the fact that my data violates this assumption?
Sorry if this is stupid question. Thank you.


do I even need to check it for normality as CLT states that if sample is big enough, our data approximates normal distribution?


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*It is wrong that, if the sample is big enough the distribution of the data/population approaches a normal distribution.
Instead, the CLT relates to (the limit of) the mean of samples (or other types of sums of variables).


*But you are right that the sample distribution of the test statistic, which is used to estimate parameters of the population distribution or estimate the error/variance, will often approach a normal distribution independent from the underlying distribution of the population.
So for large sample sizes the violation of the assumption that the error distribution is a normal distribution becomes less of a problem. (Ironically the test of normality becomes more powerful and likely to reject the hypothesis of the assumption)
