# Computing unbiased estimators of $\sigma$ for m samples of size n

Three unbiased estimators of standard deviation have been presented in the literature for m samples of size n. The first one is based on the sample ranges and can be obtained by

$\hat \sigma_1 = \frac{\bar R}{d_2}$

The second one is based on sample standard deviations and can be calculated by

$\hat \sigma_2 = \frac{\bar S}{c_4}$

The third estimator is due to sample variance and can be estimated as

$\hat \sigma_3 = \frac{\sqrt{\bar {S^2}}}{k_4}$

The denominators are constant values dependent on the sample size n and make the estimators unbiased. Does anybody know of a method to get their value in R (instead of using tables in statistical quality control books)?

• Don't these become unbiased only in conjunction with additional distributional assumptions? For example, the relationship between the range and the SD has to depend on the shape of the distribution. In that context, it's worth observing that many more unbiased estimators of SD have been studied, such as those based on linear combinations of quantiles and ranks.
– whuber
Oct 22 '10 at 15:06
• Yes, indeed. Under the normality assumption. Oct 23 '10 at 7:21

Ok, I finally found them here

• > Can you add a link (paper, explanation, hypothesis,...) to the denominator used in the first one ? (i.e. your $\hat{\sigma}_1$) - thanks. Oct 22 '10 at 14:34
• 1. Quality Control and Industrial Statistics - Duncan 2. Statistical Quality Control - Grant, Leavenworth 3. Introduction to Statistical Quality Control - Montgomery Oct 23 '10 at 8:51