Methods for Estimating the Standard Deviation

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 $\sigma$ .

Default Method

The default estimate for $\sigma$ is

$\hat{\sigma} = \frac{R_{1}/d_{2}(n_{1})+ ... + R_{N}/d_{2}(n_{N})} N$

where N is the number of subgroups for which $n_i \geq 2$ , and R_i is the sample range of the observations x_i1, . . . , $x_{in_{i}}$ in the i^th subgroup.

$R_{i} = \max_{1 \leq j \leq n_{i}}(x_{ij}) - \min_{1 \leq j \leq n_{i}}(x_{ij})$

A subgroup range R_i is included in the calculation only if $n_i \geq 2$ .The unbiasing factor d₂(n_i) is defined so that, if the observations are normally distributed, the expected value of R_i is $d_{2}(n_i)\sigma$ .Thus, $\hat{\sigma}$ is the unweighted average of N unbiased estimates of $\sigma$ .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 $\sigma$ . Refer to Burr (1969, 1976) and Nelson (1989, 1994). The MVLUE is a weighted average of N unbiased estimates of $\sigma$ of the form R_i/d₂(n_i), and it is computed as

$\hat{\sigma} = \frac{f_{1}R_{1}/d_{2}(n_{1})+ ... + f_{N}R_{N}/d_{2}(n_{N})} {f_1 + ... + f_N}$

where

f_i = [([d₂(n_i)]²)/([d₃(n_i)]²)]

A subgroup range R_i is included in the calculation only if $n_i \geq 2$ , and N is the number of subgroups for which $n_i \geq 2$ . The unbiasing factor d₃(n_i) is defined so that, if the observations are normally distributed, the expected value of $\sigma_{R_{i}}$ is $d_{3}(n_i)\sigma$ .The MVLUE assigns greater weight to estimates of $\sigma$ 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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