Methods for Estimating the Standard Deviation
When control limits are computed from the input data, three
methods (referred to as default, MVLUE, and RMSDF) are available
for estimating the process standard deviation .The method depends on whether
you specify the STDDEVIATIONS option. If you specify
this option, is estimated using subgroup
standard deviations; otherwise, is estimated using
subgroup ranges.
For an illustration of the methods, see Example 42.2.
Default Method Based on Subgroup Ranges
If you do not specify the STDDEVIATIONS option,
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).
Default Method Based on Subgroup Standard Deviations
If you specify the STDDEVIATIONS option,
the default estimate for is
where N is the number of subgroups for which
, s_{i} is the
sample standard deviation of the
i^{ th} subgroup
and
Here denotes the gamma function, and denotes
the i^{ th} subgroup mean. A subgroup standard deviation s_{i} is included
in the calculation only if . If the observations are
normally distributed, the expected value of s_{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 Based on Subgroup Ranges
If you do not specify the STDDEVIATIONS option and 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.
MVLUE Method Based on Subgroup Standard Deviations
If you specify the STDDEVIATIONS option and
SMETHOD=MVLUE,
a minimum variance linear unbiased estimate (MVLUE)
is computed for
. Refer to Burr (1969, 1976) and Nelson (1989, 1994).
This estimate is a weighted average of N unbiased estimates
of of the form s_{i}/c_{4}(n_{i}), and it is computed as
where

h_{i} = [([c_{4}(n_{i})]^{2})/(1  [c_{4}(n_{i})]^{2})]
A subgroup standard deviation s_{i}
is included in the calculation only
if , and N is the number of subgroups
for which .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.
RMSDF Method Based on Subgroup Standard Deviations
If you specify the STDDEVIATIONS option and SMETHOD=RMSDF,
a weighted rootmeansquare estimate is computed for .
The weights
are the degrees of freedom n_{i}  1.
A subgroup standard deviation s_{i} is included in the
calculation only if , and N is the number of
subgroups for which .
If the unknown standard deviation is constant across
subgroups, the rootmeansquare estimate is more efficient
than the minimum variance linear unbiased estimate. However,
in process control applications, it is generally not assumed that
is constant, and if
varies across subgroups, the rootmeansquare estimate tends
to be more inflated than the MVLUE.
Default Method Based on Individual Measurements
When each subgroup sample contains a single observation
(),
the process standard deviation is estimated as
,where is the average of the moving ranges
of consecutive measurements taken in pairs.
This is the method used to estimate for
individual measurements and moving range charts.
See
"Methods for Estimating the Standard Deviation" in Chapter 34, "IRCHART Statement."
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.