Question

How do you interpret the idea that the average has a normal distribution?

Answer 1

If take some random sample from any population although it does not have normal distribution or it has any other distribution, then the sample means or average follows a normal distribution. Suppose, we find out sample means for all possible samples with some sample size given, then the distribution of these sample means is approximately normal distribution.

In simple words, the average of the samples of same size taken from the same population with unknown distribution will follow an approximate normal distribution. This is called as the sampling distribution. Also, any sample statistic other than average follows an approximate normal distribution.