PROC SHEWHART: Constructing Charts for Standard Deviations

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Constructing Charts for Standard Deviations

The following notation is used in this section:

$\text{[math]}$

process standard deviation (standard deviation of the population of measurements)

$\text{[math]}$

standard deviation of measurements in $\text{[math]}$ th subgroup

$\text{[math]}$

sample size of $\text{[math]}$ th subgroup

$\text{[math]}$

expected value of the standard deviation of $\text{[math]}$ independent normally distributed variables with unit standard deviation

$\text{[math]}$

standard error of the standard deviation of $\text{[math]}$ independent observations from a normal population with unit standard deviation

$\text{[math]}$

100 $\text{[math]}$ th percentile $\text{[math]}$ of the $\text{[math]}$ distribution with $\text{[math]}$ degrees of freedom

Plotted Points

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

$\text{[math]}$

Central Line

By default, the central line for the $\text{[math]}$ th subgroup indicates an estimate for the expected value of $\text{[math]}$ , which is computed as $\text{[math]}$ , where $\text{[math]}$ is an estimate of $\text{[math]}$ . If you specify a known value ( $\text{[math]}$ ) for $\text{[math]}$ , the central line indicates the value of $\text{[math]}$ . Note that the central line varies with $\text{[math]}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple ( $\text{[math]}$ ) of the standard error of $\text{[math]}$ above and below the central line. The default limits are computed with $\text{[math]}$ (these are referred to as $\text{[math]}$ limits).
as probability limits defined in terms of $\text{[math]}$ , a specified probability that $\text{[math]}$ exceeds the limits

The following table provides the formulas for the limits:

Table 15.54 Limits for $\text{[math]}$ Charts
Control Limits
LCL $\text{[math]}$ lower limit = $\text{[math]}$
UCL $\text{[math]}$ upper limit = $\text{[math]}$
Probability Limits
LCL $\text{[math]}$ lower limit = $\text{[math]}$
UCL $\text{[math]}$ upper limit = $\text{[math]}$

The formulas assume that the data are normally distributed. If a standard value $\text{[math]}$ is available for $\text{[math]}$ , replace $\text{[math]}$ with $\text{[math]}$ in Table 15.54. Note that the upper and lower limits vary with $\text{[math]}$ and that the probability limits are asymmetric around the central line.

You can specify parameters for the limits as follows:

Specify $\text{[math]}$ with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
Specify $\text{[math]}$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $\text{[math]}$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\text{[math]}$ 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