How noisy is the sample standard deviation?

Question

For i.i.d. \mathcal{N}(\mu, \sigma^2) data, the sample variance satisfies \operatorname{Var}(S^2) = 2\sigma^4/(n-1). In practice you report the standard deviation (volatility), S = \sqrt{S^2}, not the variance. Derive the approximate variance and standard error of S, and give the relative error at n = 100.

Accepted Answer

Let g(v) = \sqrt{v}, so g'(v) = \frac{1}{2\sqrt{v}} and g'(\sigma^2) = \frac{1}{2\sigma}. The delta method gives \operatorname{Var}(S) \approx \left(\frac{1}{2\sigma}\right)^2 \operatorname{Var}(S^2) = \frac{1}{4\sigma^2}\cdot\frac{2\sigma^4}{n-1} = \frac{\sigma^2}{2(n-1)}. So the standard error of the volatility estimate is \operatorname{SE}(S) \approx \frac{\sigma}{\sqrt{2(n-1)}}, and the relative error is \frac{\operatorname{SE}(S)}{\sigma} \approx \frac{1}{\sqrt{2(n-1)}} \approx \frac{1}{\sqrt{2n}}. At n = 100: relative error \approx \frac{1}{\sqrt{200}} = \frac{1}{14.14} \approx 0.071, a…

How noisy is the sample standard deviation?

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