# Uncertainty of Sample skew and kurtosis

## Main Question or Discussion Point

What is the uncertainty of a samples skewness and kurtosis? Such as the uncertainty of the standard deviation is SD/sqrt(2*(N-1)). I was able to find what someone is calling the Standard Error of these but they both only depend on N which doesn't make sense to me.

Skewness Standard Error: sqrt((N^2-1)/((N-3)*(N+5)))

Kurtosis Standard Error: SSE*sqrt(6*N*(N-1)/((N-2)*(N+1)*(N+3)))

These get close to 1 in the range of measurements that I'm looking at which doesn't have the same behavior of the uncertainty of the standard deviation, which goes from inf to 0 with increasing measurements. And it doesn't depend on the value of the skew or kurtosis...

## Answers and Replies

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Turns out the terms I were looking for are either the variance of sample skewness and kurtosis, or standard deviation of skewness or kurtosis. I have found the following:

Std. Dev. of Skewness: sqrt((6*(N-2)*(Std. Dev.)^2)/((N+1)*(N+3)))
Std. Dev. of Kurtosis: sqrt((24*N*(N-2)*(N-3)*(Std. Dev.)^2)/((N+1)^2*(N+3)*(N+5)));

I hope these are correct as they seem rather obscure (which I can see by the fact no one answered) though I'd think they should be much more well known.

Stephen Tashi
You should clarify whether you are asking about statistics of a population or about estimators of those statistics. For example, in the formula you gave for the standard deviation of the sample skewness, you'd have to know the population standard deviation exactly in order to apply it. If you don't know the statistics of the population exactly and only have sample data then all you know is the sample standard deviation.