Estimating error in 3sigma?

Question

I am designing a process, which etches some holes into a substrate. The goal of the process is to reduce the population standard deviation of the hole dimensions.

I recently collected 1000 hole measurements, to estimate the population / sample standard deviation, and the error associated with the standard deviation. The reason we are looking at the error associated with the standard deviation is because we expect that the signal in subsequent processes might be of the same order as that of the error in the sample standard deviation. So we want to quantify the sample standard deviation. We calculated the sample mean and sample standard deviation from this sample. I understand the standard error of the sample mean is given by $\sigma/\sqrt N$.

However, we don't want to collect 1000 points every time, because its quite expensive. So I want to obtain a theoretical plot of the error of the sample standard deviation, as a function of the number of holes measured in our experiment. Our goal is to minimize the standard deviation of the population standard deviation for this process. So I am trying to estimate the error of the sample standard deviation, so that we can then decide how many points we can collect, and yet have enough signal to noise ratio, so that our experiments have some meaning.

What formula can I use to do so?

Answer 1

The variance of $s^2_{n-1}$ is

$$\text{Var}(s^2_{n-1})=({\mu_4}-{_{n-3}\over ^{n-1}}\,\sigma^4)/n.$$

This is derived in various places -- for example, a reasonably simple derivation is given here.

This doesn't depend on the distribution of the original measurements, just on fourth and second moments. However, the variance of the standard deviation does.

Nevertheless we can approximate it, for example via Taylor series.

$t(X) = t(\mu+X-\mu) = t(\mu)+(X-\mu)t'(\mu) +\frac{(X-\mu)^2}{2!}t''(\mu) +...$

Hence $\text{Var}[t(X)] = t'(\mu)^2\text{Var}(X-\mu)+t'(\mu)t''(\mu)\text{Cov}(X-\mu,(X-\mu)^2)+ \frac{t''(\mu)^2}{4}\text{Var}((X-\mu)^2)+...$

(where $\sigma^2,\mu_3,\mu_4$ are central moments of the variable $X$).

... as long as that series converges!

Here $t(X)=X^\frac12$ so $t'(X) =\frac12X^{-\frac12}$ and $t''(X)=-\frac14X^{-\frac32}$.

Here, of course, $X=s^2_{n-1}$, not the original variable we were computing the variance of. This expansion is tedious (often people ignore all but the first term), but you can see that the first few terms involve third and fourth moments of the sample variance.

Answer 2

I think the simplest thing for you to do is just to bootstrap your standard deviation. That is, take a large number of samples, with replacement, from your sample and compute the SD on each. Then you can get the SD of that bootstrapped sampling distribution of the SD. With 1000 data, this should be sufficiently robust for your purposes. There are a lot of threads on bootstrapping on CV already; you can learn more by reading through some of them. Try clicking here: bootstrap.