Methods for Estimating the Standard Deviation
When control limits are determined from the input data, two
methods (referred to as default and MVLUE) are available
for estimating .Default Method
The default estimate for is
where N is the number of subgroups for which
, and
R_{i} is the sample range of the observations
x_{i1}, . . . , in the
i^{ th} subgroup.
A subgroup range R_{i} is included in the calculation
only if .The unbiasing factor d_{2}(n_{i}) is defined
so that, if the observations are normally distributed,
the expected value of R_{i} is .Thus, is the unweighted average of N unbiased
estimates of .This method is described in the ASTM Manual on Presentation of
Data and Control Chart Analysis (1976).
MVLUE Method
If you specify SMETHOD=MVLUE,
a minimum variance linear unbiased estimate (MVLUE) is computed for
. Refer to Burr (1969, 1976) and Nelson (1989, 1994).
The MVLUE is a weighted average of N unbiased estimates
of of the form R_{i}/d_{2}(n_{i}), and it is computed as
where

f_{i} = [([d_{2}(n_{i})]^{2})/([d_{3}(n_{i})]^{2})]
A subgroup range R_{i} is included in the calculation only
if , and N is the number of subgroups
for which . The unbiasing factor d_{3}(n_{i}) is defined
so that, if the observations are normally distributed, the expected
value of is .The MVLUE assigns greater weight to estimates of from
subgroups with larger sample sizes, and it is intended for
situations where the subgroup sample sizes vary. If the subgroup
sample sizes are constant, the MVLUE reduces to the default
estimate.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.