Quantifying Uncertainty in Small-Sample Standard Deviation Estimates
Summary
The document asks how to assess the precision of a standard deviation estimated from a small sample of returns. One answer proposes treating the estimate itself as a random quantity: repeatedly resample the observed data, recompute the standard deviation, and use the resulting distribution to form confidence intervals. This bootstrap suggestion is practical in spirit, though the answer points elsewhere for implementation details and does not spell out assumptions or resampling choices.
Another answer discusses the Berry–Esseen theorem as a bound on how quickly a sample-mean distribution approaches a normal approximation under moment conditions. That result concerns the mean and does not directly provide a confidence interval for the sample standard deviation, so its relevance to the original question is indirect. The exchange supplies no computed uncertainty interval for the example sample and no sample-size threshold for meaningful estimates. It highlights possible approaches while leaving their assumptions and suitability for return data to be examined.
Key ideas
- A bootstrap can estimate uncertainty by resampling observations and recalculating the standard deviation.
- Confidence intervals can be derived from the bootstrap distribution, subject to suitable resampling choices.
- The Berry–Esseen theorem bounds normal approximation error for sample means under moment conditions.
- A bound for the sample mean does not directly quantify uncertainty in a sample standard deviation.
- The discussion gives no sample-size rule or worked uncertainty interval for the example.
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Full text
# What is the precision of standard deviation estimates with small samples?
# What is the precision of standard deviation estimates with small samples?
I was asked today to "quantify" the precision of an estimated the standard deviation from a small sample, I was not sure how to answer.
The case is quite simple, I have a sample of $n=25$ measures (returns as you would have guessed). I used the classic unbiased estimator for the standard deviation:
$$\sigma_x = \sqrt{\frac{1}{N-1}\sum_{n=1}^n (x_i-\bar{x})^2}$$
The underlying question was : how much data do we need for the standard deviation to be statistically meaningful.
I read here that computing the standard error of the standard deviation is difficult to estimate, but I wanted to know if there was a common procedure used by you guys in general?
## Answer by Ram Ahluwalia (score 10, accepted)
https://quant.stackexchange.com/a/2525
Treat the estimate of standard deviation as a random variable. Then you can bootstap the sample estimate and generate t-statistics and associated confidence intervals for your statistics. I describe a generic boostrap process on this post.
## Answer by Beer4All (score 5)
https://quant.stackexchange.com/a/2524
Actually you should be interested by the Berry Essen's theorem which precises the rate of convergence of the central limit theorem.
Given i.i.d. $X_1,\dots, X_n \sim X$
1) GLN : assuming $E(X)<\infty$ then $\overline{X}_n-E(X)\to 0 $
2) CLT ("rate" of the GLN) : assuming $E(X^2)<\infty$ then $\frac{\sqrt{n}}{\sigma^2} \big(\overline{X}_n-E(X)\big)\to N(0,1) $
3) Berry Essen ("rate" of the CLT) : assuming $E(X^3)<\infty$ , then
$\sup_{x\in \mathbb{R}}\bigg| \,F_{\frac{\sqrt{n}}{\sigma^2} \big(\overline{X}_n-E(X)\big)}(x) - F_{N_{0,1}}(x) \bigg| \leq \frac{0.34445 E|X|^3 + 0.16844}{\sqrt{n}}$
Where $F_{}$ holds for the CDF.
This is an upper bound (of the order $\sqrt{n}$) usable for your CLT approximation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.