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 XRCHART Statement

## 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 Ri is the sample range of the observations xi1, . . . , in the i th subgroup.
A subgroup range Ri is included in the calculation only if .The unbiasing factor d2(ni) is defined so that, if the observations are normally distributed, the expected value of Ri 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 Ri/d2(ni), and it is computed as
where
fi = [([d2(ni)]2)/([d3(ni)]2)]

A subgroup range Ri is included in the calculation only if , and N is the number of subgroups for which . The unbiasing factor d3(ni) 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.

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