If one is dealing with small sample sizes, let's say $8-16$ observations per sample, and we are interested in estimates of the standard-deviation (let us also assume Gaussian statistics), is there a reason not to apply sample-size bias correction?
For example the arithmetic sample standard deviation, $$s = \sqrt{\frac{1}{N-1} \sum_{i}^{N} {(x_{i} - \bar{x})^2}} \text{,} $$ is a biased estimator -- especially for small sample sizes. This can be corrected with the $c_4(N)$ bias correction factor. This gives an unbiased estimate of the standard deviation as $$\sigma_{\rm{est}} = s /c_{4}(N) \text{.}$$
There are bias correction factors for small-sample size estimates attained from MLE methods, or the $\rm{MAD}$, $Q_{n}$, and $S_{n}$ dispersion estimators which can also be used to make estimates on the standard deviation.
I ask because if we use an estimate of the standard-deviation to produce a standard error for the mean, i.e. $$\rm{S.E} = \sigma_{\rm{est}}/\sqrt{N}$$ for small sample sizes the correction factor can make quite a difference in the result, so:
- Is there a good reason not to account for sample sizes, especially when they are small?
- Does including bias correction for estimates which are used to produce standard errors and/or weights, have any undesirable consequences?