Methods for Estimating the Standard Deviation

XCHART Statement

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 $\sigma$ .The method depends on whether you specify the STDDEVIATIONS option. If you specify this option, $\sigma$ is estimated using subgroup standard deviations; otherwise, $\sigma$ 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 $\sigma$ 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).

Default Method Based on Subgroup Standard Deviations

If you specify the STDDEVIATIONS option, the default estimate for $\sigma$ is

$\hat{\sigma} = \frac{s_{1}/c_{4}(n_{1})+ ... + s_{N}/c_{4}(n_{N})} N$

where N is the number of subgroups for which $n_i \geq 2$ , s_i is the sample standard deviation of the i^th subgroup

$s_{i} = \sqrt{ \frac{1}{n_{i} - 1} \sum^{n_i}_{j=1}(x_{ij}-\bar{X}_{i})^2}$

and

$c_{4}(n_{i}) = \frac{\Gamma(n_{i}/2)\sqrt{2/(n_{i}-1)}} {\Gamma((n_{i}-1)/2)}$

Here $\Gamma(\cdot)$ denotes the gamma function, and $\bar{X}_{i}$ denotes the i^th subgroup mean. A subgroup standard deviation s_i is included in the calculation only if $n_i \geq 2$ . If the observations are normally distributed, the expected value of s_i is $c_{4}(n_{i})\sigma$ .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 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 $\sigma$ . 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.

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 $\sigma$ . Refer to Burr (1969, 1976) and Nelson (1989, 1994). This estimate is a weighted average of N unbiased estimates of $\sigma$ of the form s_i/c₄(n_i), and it is computed as

$\hat{\sigma} = \frac{h_{1}s_{1}/c_{4}(n_{1})+ ... + h_{N}s_{N}/c_{4}(n_{N})} {h_1 + ... + h_N}$

where

h_i = [([c₄(n_i)]²)/(1 - [c₄(n_i)]²)]

A subgroup standard deviation s_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 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.

RMSDF Method Based on Subgroup Standard Deviations

If you specify the STDDEVIATIONS option and SMETHOD=RMSDF, a weighted root-mean-square estimate is computed for $\sigma$ .

$\hat{\sigma} = \frac{\sqrt{(n_{1} - 1)s_1^2 + ... + (n_{N} - 1)s_{N}^2}} {c_{4}(n)\sqrt{n_{1} + ... + n_{N} - N}}$

The weights are the degrees of freedom n_i - 1. A subgroup standard deviation s_i is included in the calculation only if $n_i \geq 2$ , and N is the number of subgroups for which $n_i \geq 2$ .

If the unknown standard deviation $\sigma$ is constant across subgroups, the root-mean-square estimate is more efficient than the minimum variance linear unbiased estimate. However, in process control applications, it is generally not assumed that $\sigma$ is constant, and if $\sigma$ varies across subgroups, the root-mean-square estimate tends to be more inflated than the MVLUE.

Default Method Based on Individual Measurements

When each subgroup sample contains a single observation ( $n_{i} \equiv 1$ ), the process standard deviation $\sigma$ is estimated as $\hat{\sigma} = \bar{R}/d_{2}(2)$ ,where $\bar{R}$ is the average of the moving ranges of consecutive measurements taken in pairs. This is the method used to estimate $\sigma$ for individual measurements and moving range charts. See "Methods for Estimating the Standard Deviation" in Chapter 34, "IRCHART Statement."

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