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For a data set that is Gaussian distributed, $\sigma$ defines the standard-deviation of the distribution. My question is, what is the correct indention or terminology for the following: $$\frac{\sigma}{\sqrt{2N-2}} \text{.}$$
My interpretation is that it is the "error" in the standard deviation, $\sigma$. Is this a correct way to view this?
It's important to distinguish between the true population value of a parameter of a statistical distribution and the estimate of that value obtained from a sample of size $N$. The symbol $\sigma$ is conventionally used to represent the population standard deviation, the square root of the population variance $\sigma^2$. A sample estimate of the variance is conventionally represented as $s^2$ and a sample estimate of the standard deviation as $s$.
This question shows the formula for the standard error of estimates of the variance, $s^2$, when sampling from a normal distribution. The answer to that question shows a way to convert such standard error estimates of $s^2$ to standard error estimates of $s$ that holds in the limit of large sample sizes for sampling from any of a large class of distributions. A bit of math shows that if you replace $\sigma$ in your formula with $s$ you have the corresponding formula for the standard error of the sample standard deviation in the case of a normal distribution.
