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XCHART Statement |
This section provides formulas for various methods used to estimate the standard deviation . One method is applicable with individual measurements, and three are applicable with subgrouped data. The methods can be requested with the SMETHOD= option.
Note that you can compute alternative estimates (for instance, robust estimates or estimates based on variance components models) by analyzing the data with SAS modeling procedures or your own DATA step program. Such estimates can be passed to the CUSUM procedure as values of the variable _STDDEV_ in a LIMITS= data set.
where n_{i} is the sample size of the i^{ th} subgroup, N is the number of subgroups for which , s_{i} is the sample standard deviation of the observations x_{i1}, ... ,x_{ini} in the i^{ th} subgroup.
s_{i} | is the standard deviation of the i^{ th} subgroup. |
c_{4}(n_{i}) | is the unbiasing factor defined previously. |
n_{i} | is the i^{ th} subgroup sample size, i = 1,2, ... ,N. |
N | is the number of subgroups for which . |
The estimate is
The MVLUE assigns greater weight to estimates of from subgroups with larger sample sizes and is intended for situations where the subgroup sample sizes vary. If the subgroup sample sizes are constant, the MVLUE reduces to the default estimate (NOWEIGHT).
where
n_{i} | is the sample size of the i^{ th} subgroup. |
N | is the number of subgroups for which . |
s_{i} | is the sample standard deviation of the i^{ th} subgroup. |
c_{4}(n_{i}) | is the unbiasing factor defined previously. |
n | is equal to (n_{1}+ ... +n_{N})-(N-1) . |
The weights in the root-mean-square expression are the degrees of freedom n_{i}-1. A subgroup standard deviation s_{i} is included in the calculation only if .
If the unknown standard deviation is constant across subgroups, the root-mean-square estimate is more efficient than the minimum variance linear unbiased estimate. However, as noted by Burr (1969), "the constancy of is the very thing under test," and if varies across subgroups, the root-mean-square estimate tends to be more inflated than the MVLUE.
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