Methods for Estimating the Standard Deviation :: SAS/QC(R) 13.1 User's Guide

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

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

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 ith 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 $d_{2}(n_ i)$ 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 American Society for Testing and Materials (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 $R_ i/d_2(n_ i)$ , and it is computed as

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

where

$f_ i = \frac{[d_2(n_ i)]^{2}}{[d_3(n_ i)]^{2}}$

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 $d_3(n_ i)$ 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.

RCHART Statement: SHEWHART Procedure

Methods for Estimating the Standard Deviation

Default Method

MVLUE Method