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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?

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    $\begingroup$ 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 the CLT states and means 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). $\endgroup$ – whuber Mar 17 '17 at 13:01
  • $\begingroup$ sampling generally assumes unlimited samples , only sample size n is generally called out -- so this is what may have confused you in my question. So ignore that. Just say that the sample size n is large enough for clt to kick in if that was the condition for clt to kick in, but I don't know if it will kick in because it doesn't say anything outside sample mean. Thanks $\endgroup$ – Dhiraj Mar 17 '17 at 17:07
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    $\begingroup$ Technically CLT never "kicks in" -- for finite sample sizes, it is always only approximately true. $\endgroup$ – Chill2Macht Mar 18 '17 at 16:58
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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

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