Is the sampling distribution for skewness and kurtosis normal?

[dhiraj]

The Central Limit Theorem specifically calls out a sample mean to be following normal distribution (for \(\displaystyle n>=30\) ), But I am referring to certain text, and it is calculating \(\displaystyle z\) values for sample \(\displaystyle skewness\) and sample \(\displaystyle kurtosis\) assuming that these follow Normal distribution. Is it correct?

In short, if we take unlimited number of samples each of size \(\displaystyle n\) (where \(\displaystyle n>=30\) ) then for each sample \(\displaystyle skewness\) and \(\displaystyle kurtosis\) will vary. So these are random variables. The question is -- Is the sampling distribution of these random variables Normal?. And if it is, what is the mean \(\displaystyle \mu\) and standard deviation \(\displaystyle \sigma\) for that?

 

[jvicens]

I can confirm that the Central Limit Theorem states that for a large enough sample size (n>=30), the sample mean will follow a normal distribution. This means that if we were to take an infinite number of samples, the distribution of their means would be normal.

However, the Central Limit Theorem does not apply to other sample statistics such as skewness and kurtosis. These statistics are not measures of the sample mean, but rather measures of the shape of the data distribution. Therefore, their sampling distribution may not necessarily follow a normal distribution.

It is important to note that the distribution of sample skewness and kurtosis will vary for each sample, but as the sample size increases, the distribution will tend towards a normal distribution. This is known as the asymptotic normality of these statistics.

In regards to calculating z-values for sample skewness and kurtosis, it is not incorrect to assume that they follow a normal distribution. However, this assumption may not hold for smaller sample sizes. As for the mean and standard deviation of the sampling distribution of these statistics, they can be estimated using mathematical formulas, but they may also vary depending on the underlying distribution of the data.

In conclusion, while the Central Limit Theorem does not directly apply to sample skewness and kurtosis, their sampling distribution may still approach a normal distribution as the sample size increases. However, it is important to consider the underlying distribution of the data when making assumptions about these statistics.