PROC SHEWHART: Constructing Charts for Individual Measurements and Moving Ranges

The SHEWHART Procedure

Constructing Charts for Individual Measurements and Moving Ranges

The following notation is used in this section:

$\text{[math]}$

process mean (expected value of the population of measurements)

$\text{[math]}$

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

$\text{[math]}$

the $\text{[math]}$ th individual measurement

$\text{[math]}$

mean of the individual measurements, computed as $\text{[math]}$ , where $\text{[math]}$ is the number of individual measurements

$\text{[math]}$

number of consecutive measurements used to calculate the moving ranges (by default, $\text{[math]}$ )

$\text{[math]}$

moving range computed for the $\text{[math]}$ th subgroup (corresponding to the $\text{[math]}$ th individual measurement). If $\text{[math]}$ , then $\text{[math]}$ is assigned a missing value. Otherwise,

$\text{[math]}$

This formula assumes that $\text{[math]}$ are nonmissing.

$\text{[math]}$

average of the nonmissing moving ranges, computed as

$\text{[math]}$

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

$\text{[math]}$

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

$\text{[math]}$

100 $\text{[math]}$ th percentile $\text{[math]}$ of the standard normal distribution

$\text{[math]}$

100 $\text{[math]}$ th percentile $\text{[math]}$ of the distribution of the range of $\text{[math]}$ independent observations from a normal population with unit standard deviation

Plotted Points

Each point on an individual measurements chart, indicates the value of a measurement ( $\text{[math]}$ ).

Each point on a moving range chart indicates the value of a moving range ( $\text{[math]}$ ). With $\text{[math]}$ , for example, if the first three measurements are 3.4, 3.7, and 3.6, the first moving range is missing, the second moving range is $\text{[math]}$ , and the third moving range is $\text{[math]}$ .

Central Lines

By default, the central line on an individual measurements chart indicates an estimate for $\text{[math]}$ , which is computed as $\text{[math]}$ . If you specify a known value ( $\text{[math]}$ ) for $\text{[math]}$ , the central line indicates the value of $\text{[math]}$ .

The central line on a moving range chart indicates an estimate for the expected moving range, computed as $\text{[math]}$ where $\text{[math]}$ . If you specify a known value ( $\text{[math]}$ ) for $\text{[math]}$ , the central line indicates the value of $\text{[math]}$ .

Control Limits

You can compute the limits

as a specified multiple ( $\text{[math]}$ ) of the standard errors of $\text{[math]}$ and $\text{[math]}$ 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 $\text{[math]}$ or $\text{[math]}$ exceeds the limits

The following table provides the formulas for the limits:

Table 13.20 Limits for Individual Measurements and Moving Range Charts
Control Limits
Individual Measurements Chart	LCL $\text{[math]}$ lower control limit = $\text{[math]}$
	UCL $\text{[math]}$ upper control limit = $\text{[math]}$
Moving Range Chart	LCL $\text{[math]}$ lower control limit = $\text{[math]}$
	UCL $\text{[math]}$ upper control limit = $\text{[math]}$

Probability Limits
Individual Measurements Chart	LCL $\text{[math]}$ lower control limit = $\text{[math]}$
	UCL $\text{[math]}$ upper control limit = $\text{[math]}$
Moving Range Chart	LCL $\text{[math]}$ lower control limit = $\text{[math]}$
	UCL $\text{[math]}$ upper control limit = $\text{[math]}$

The formulas assume that the measurements are normally distributed. Note that the probability limits for the moving range are asymmetric about the central line. If standard values $\text{[math]}$ and $\text{[math]}$ are available for $\text{[math]}$ and $\text{[math]}$ , replace $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$ with $\text{[math]}$ in Table 13.20.

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 $\text{[math]}$ with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\text{[math]}$ with the MU0= option or with the variable _MEAN_ in the LIMITS= data set.
Specify $\text{[math]}$ with the SIGMA0= option or with the variable _STDDEV_ in the LIMITS= data set.

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