## Bias in estimated standard deviationsThis
graphic shows the Helmert distribution for standard deviations
estimated at various sample sizes from a Normal distribution. There are
two common estimators: one uses If
there were no bias in the estimated standard deviation, the
distributions would have their mean at 1.0; the horizontal axis in the
graphic is the ratio of the estimated to known standard deviation. The
mean values are the vertical lines on the distributions; these mean
values change with the sample size. The bias becomes very small at
larger sample sizes, and the distributions become nearly Normal. It
can be argued that the bias corrections are small, but they are worth
considering at the small sample sizes used in quality control, or in
lab exercises in schools. In QC, these correction factors are important
enough to have their own symbols (c2 and c4). See, e.g., Wheeler, D.
J., Advanced Topics in Statistical Process Control, SPC Press (1995), p. 58 (ISBN 0-945320-45-0) Also see Duncan, A. J., Quality Control and Industrial Statistics, 4th Ed., Irwin (1974), p. 139 and Appendix II, Table M (ISBN 0-256-01558-9).The
idea of c2 and c4, which are obtained from the analytically-calculated
mean values (expected values) of the respective distributions, is that
the estimated s is to be divided by c2 (if n was used for s) or c4 (if n-1 was used for s) in order to obtain an unbiased estimate of the population standard deviation.Note that at small sample sizes there is a lot of variability in the values of It
should also be noted that the Helmert distribution is closely related
to the "Chi" distribution; see, e.g., Johnson and Kotz, Press the small button at the lower left to sweep the sample size automatically.
WCEvans (9/09), Created with GeoGebra |