# Standard error of standard deviation, skewness and kurtosis

I want to use this formula (shown below) for my work (not math based) to calculate the uncertainty in the sample standard deviation (obtained from the link below): Calculating uncertainty in standard deviation

$$SE(\sigma) = \frac{\sigma}{\sqrt(2N-2)}$$

where $$N$$ is the size of the sample record.

However I cannot find any statistics books which cover this in anywhere? They all cover the standard error of the mean. The first sentence of the wikipedia article suggests that it should be possible to obtain a standard error for the standard deviation:

The standard error (SE) of a statistic (usually an estimate of a parameter)
is the standard deviation of its sampling distribution[1] or an estimate of
that standard deviation. If the parameter or the statistic is the mean, it
is called the standard error of the mean (SEM).


Can anyone show me how this formula is derived and any simple(!) books which at least mention it?

Also, can a similar formula be found for standard error in kurtosis and skewness?

PS: I am not yet able to make a comment on the original thread. I made a previous question but it was asking too much so I'm asking again (with less stuff).

• If you have a large data set of approximately independent and identically distributed experiments, then you can appeal to the CLT and Cochran's theorem that the sample variance, $S^2$, follows a scaled chi-square distribution, i.e. $(n-1)S^2/\sigma^2\sim \chi_{n-1}^2$ (where $\sigma^2$ is the true variance). You can then use this to obtain confidence intervals on the variance $\sigma^2$ in terms of the point estimate, and hence also for standard deviation. See, en.wikipedia.org/wiki/… – Nap D. Lover Jan 18 at 0:38
• Hi, yes, I have used this methodology to obtain confidence intervals already. But it is not trivial to go from the confidence interval to obtaining the absolute uncertainty as it is the case for confidence intervals for the mean. – slew123 Jan 18 at 0:41
• If, for example, you look at: en.wikipedia.org/wiki/Confidence_interval#Basic_steps you can see that the confidence interval for the true mean is simply the sample mean plus/minus the standard error multiplied by its confidence level, C. You can then estimate the percentage uncertainty using this standard error. However the confidence interval of the standard deviation has no such quantity which you can use to obtain a percentage uncertainty. That is why I wanted to use the standard error of the standard deviation. – slew123 Jan 18 at 0:47
• Let me see if I understand this correctly: in the case of the mean, by "percentage uncertainty using the fractional error" you are just referring to taking the standard error, $z^* \cdot (\sigma/\sqrt{n})$ divided by the point estimate, $\bar{x}$? Then, you wish for something analogous for the case of estimating standard deviation? If so, my comment, indeed, is not too useful, sorry for the noise. – Nap D. Lover Jan 18 at 0:54
• Or perhaps the other way around (the point estimate divided by the standard error of the point estimate). – Nap D. Lover Jan 18 at 1:00