# Is the sampling distribution for skewness and kurtosis normal?

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?

• 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).
– whuber
Mar 17, 2017 at 13:01
• 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 Mar 17, 2017 at 17:07
• Technically CLT never "kicks in" -- for finite sample sizes, it is always only approximately true. Mar 18, 2017 at 16:58

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