# Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study?

In a simulation study, is there any difference between

$\bullet$ estimating the variance $\sigma^2$, $1000$ times and taking its average, and

$\bullet$ estimating the standard deviation $\sigma$, $1000$ times and taking its average?

Can I do anyone of these? Is there any preference of doing a particular one?

• Clearly there's some differences because the variance and the standard deviation are not the same. Can you be more specific about what you're after? – Glen_b Feb 19 '17 at 10:08
• We prefer the variance because the formula for variance is unbiased for any underlying distribution. You may find the answers to your question on this page stats.stackexchange.com/questions/249688/… – Hugh Feb 19 '17 at 10:10
• @hugh are you sure unbiasedness should be the only criterion? – Glen_b Feb 19 '17 at 10:11
• @Glen_b In this link bmcmedresmethodol.biomedcentral.com/articles/10.1186/… (Table 1), I am not understanding why did authors estimate $\sigma_0$, $\sigma_1$ instead of $\sigma_0^2$, $\sigma_1^2$? – user81411 Feb 19 '17 at 10:20
• Also joophox.net/publist/methodology05.pdf, authors estimated $\sigma.$ – user81411 Feb 19 '17 at 10:22

## 1 Answer

I find this question of interest because it highlights the artificial nature of seeking unbiasedness above everything else. A few points:

• the variance $\sigma^2$ allows for an unbiased estimator, while the square root of that estimator $\hat\sigma_n$ is biased [by Jensen's inequality];

• there is no generic unbiased estimator of $\sigma$ [generic meaning across all distributions];

• for a scale or location-scale family of distributions, since $\sigma$ is a scale, the expectation $\mathbb{E}^P[\hat\sigma_n]$ can be written as $$\mathbb{E}^P[\hat\sigma]=c(P,n)\sigma$$ where $n$ is the sample size and $P$ is the family of distributions. Hence bias can be corrected family-wise

• I guess the real reason one often is pooling variance (and not standard deviation) is that (with normal distributed data) it leads to cleaner distribution theory (F-dist). – kjetil b halvorsen Feb 25 '17 at 15:40