Standard Deviation of Standard Deviation

Standard deviation is random, or more specifically, sample standard deviation $s$ is random. This randomness is inherent when randomly sampling a population, because each sample will contain slightly different data.

This spread is quantified using standard error $SE$ , which is equal to standard deviation of a statistic’s sampling distribution. Standard error is commonly used with sample mean $\hat{x}$ , but the same concept applies to standard deviation. In other words, the standard deviation of standard deviation, is the standard error of sample standard deviation ${SE}(S)$

The following is a basic standard error of sample standard deviation calculator:

Standard Deviation

\sigma \ or \ s =

Sample Size

n =

Standard Error of Sample Standard Deviation

SE(S) \ or \ \hat{SE}(S) =

0.131

Standard Error of Sample Variance

SE(S^{2}) \ or \ \hat{SE}(S^{2}) =

0.263

This blog post is based on the formulas in: Standard Errors of Mean, Variance, and Standard Deviation Estimators, Sangtae Ahn and Jeffrey A. Fessler, University of Michigan, 2003

Standard Error of Sample Standard Deviation

SE(S) = \frac{\sigma}{\sqrt{2(n-1)}}

\hat{SE}(S) = \frac{s}{\sqrt{2(n-1)}}

$\sigma$ is the population standard deviation
$s$ is the sample standard deviation
$SE(S)$ is the standard error of standard deviation
$\hat{SE}(S)$ is the estimator of standard error of standard deviation
$n$ is the sample size and $n-1$ is the degrees of freedom

The difference between $SE(S)$ and $\hat{SE}(S)$ is the use of using population standard deviation $\sigma$ , however in most practical situations, the population is unknown and $\hat{SE}(S)$ will need to be used.

It is important to note that the standard error of standard deviation is not the square root of the standard error of sample variance, $\sqrt{SE(S^2)} \neq SE(S)$ . See next section.

Standard Error of Sample Variance

SE(S^2) = \sigma^2\sqrt{\frac{2}{n-1}}

\hat{SE}(S^2) = s^2\sqrt{\frac{2}{n-1}}

$\sigma^2$ is the population variance
$s^2$ is the sample variance
$SE(S^2)$ is the standard error of sample variance
$\hat{SE}(S^2)$ is the estimator of standard error of sample variance
$n-1$ is the degrees of freedom

The distribution of sample variance follows a chi squared distribution with $n-1$ degrees of freedom $\chi^{2}_{n-1}$

Sample Standard Deviation Demo

Normal PDF

N(0,1)

Distribution of Sample Stdev

s

Sample more data to see distribution

Sample Size

n =

Standard Error of Sample Standard Deviation

SE(S)=

0.236

Estimated Standard Error of Sample Standard Deviation (using replicate #)

\hat{SE}(S)=

Standard Deviation of Sample Standard Deviation (using 0 replicates)

#	Size $n$	Mean $\hat{x}$	Stdev $s$	Distribution

The above demo allows you to generate repeat samples from the standard normal distribution. The standard error of standard deviation is calculated using multiple methods for comparison.

$SE(S)$ is calculated using the population standard deviation $\sigma$ and sample size $n$
$\hat{SE}(S)$ is calculated using the sample standard deviation $s$ of a single generated replicate (only the most recent one)
It is also possible to calculate standard deviation of standard deviation using all of the replicates, effectively treating it like a sample. This would converge with the $SE(S)$ as more replicates are added.