The Central Limit Theorem specifically calls out a sample mean to be following a normal distribution (for $n \ge 30$ ). But I am referring to a certain text, and it is calculating $z$ values for sample $skewness$ and sample $kurtosis$ assuming that these follow a normal distribution.
Is the book correct?
In short, if we take unlimited number of samples each of size $n$ (where $n \ge 30$ ) then for each sample $skewness$ and $kurtosis$ will vary. So these are random variables.
Question: Is the sampling distribution of these random variables normal? And if it is, what is the mean $\mu$ and standard deviation $\sigma$ for that?
Best Answer
Answered in comments, copied below:
Equating "unlimited" with "$ n \ge 30$" is mistaken, as well as this statement of the CLT. Both errors might underlie the approach in the text you are using. A review of What intuitive explanation is there for the central limit theorem? would shed light on these issues. (+1) The short answer is that the sampling distributions of skewness and kurtosis are never normal, but asymptotically they become approximately normal (albeit extremely slowly). – whuber