Constructing Charts for Standard Deviations :: SAS/QC(R) 13.1 User's Guide

Constructing Charts for Standard Deviations

The following notation is used in this section:

$\sigma$	process standard deviation (standard deviation of the population of measurements)
$s_{i}$	standard deviation of measurements in ith subgroup $s_{i} = \sqrt {(1/(n_ i-1))( (x_{i1} - \bar{X_{i}})^2 + \cdots + (x_{in_{i}} - \bar{X_{i}})^2)}$
$n_{i}$	sample size of ith subgroup
$c_{4}(n)$	expected value of the standard deviation of n independent normally distributed variables with unit standard deviation
$c_{5}(n)$	standard error of the standard deviation of n independent observations from a normal population with unit standard deviation
$\chi ^{2}_{p}(n)$	100pth percentile $(0<p<1)$ of the $\chi ^{2}$ distribution with n degrees of freedom

Plotted Points

Each point on an s chart indicates the value of a subgroup standard deviation ( $s_{i}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 13, the value plotted for this subgroup is

$s_{10}= \sqrt {((12-15)^2 + (15-15)^2 + (19-15)^2 + (16-15)^2 + (13-15)^2)/4 } = 2.739$

Central Line

By default, the central line for the ith subgroup indicates an estimate for the expected value of $s_{i}$ , which is computed as $c_{4}(n_{i})\hat{\sigma }$ , where $\hat{\sigma }$ is an estimate of $\sigma$ . If you specify a known value ( $\sigma _{0}$ ) for $\sigma$ , the central line indicates the value of $c_{4}(n_{i})\sigma _{0}$ . Note that the central line varies with $n_{i}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of $s_{i}$ above and below the central line. The default limits are computed with k = 3 (these are referred to as $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that $s_{i}$ exceeds the limits

The following table provides the formulas for the limits:

Table 17.54: Limits for s Charts

Control Limits
LCL = lower limit = $\mbox{max}(c_{4}(n_{i})\hat{\sigma } - kc_{5}(n_{i})\hat{\sigma },0)$
UCL = upper limit = $c_{4}(n_{i})\hat{\sigma } + kc_{5}(n_{i})\hat{\sigma }$
Probability Limits
LCL = lower limit = $\hat{\sigma }\sqrt {\chi ^{2}_{\alpha /2}(n_ i - 1)/(n_ i-1)}$
UCL = upper limit = $\hat{\sigma }\sqrt {\chi ^{2}_{1-\alpha /2}(n_ i - 1)/(n_ i-1)}$

The formulas assume that the data are normally distributed. If a standard value $\sigma _{0}$ is available for $\sigma$ , replace $\hat{\sigma }$ with $\sigma _{0}$ in Table 17.54. Note that the upper and lower limits vary with $n_{i}$ and that the probability limits are asymmetric around the central line.

You can specify parameters for the limits as follows:

Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
Specify $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\sigma _{0}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

SCHART Statement: SHEWHART Procedure

Constructing Charts for Standard Deviations

Plotted Points

Central Line

Control Limits