PROC SHEWHART: Constructing Charts for Medians and Ranges

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Constructing Charts for Medians and 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)

$\text{[math]}$

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

$\text{[math]}$

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

$\text{[math]}$

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

$\text{[math]}$

the number of subgroups

$\text{[math]}$

$\text{[math]}$ th measurement in the $\text{[math]}$ th subgroup, $\text{[math]}$

$\text{[math]}$

$\text{[math]}$ th largest measurement in the $\text{[math]}$ th subgroup. Then

$\text{[math]}$

weighted average of subgroup means

$\text{[math]}$

median of the measurements in the $\text{[math]}$ th subgroup:

$\text{[math]}$

average of the subgroup medians:

$\text{[math]}$

median of the subgroup medians. Denote the $\text{[math]}$ th largest median by $\text{[math]}$ so that $\text{[math]}$ .

$\text{[math]}$

standard error of the median of $\text{[math]}$ independent, normally distributed variables with unit standard deviation (the value of $\text{[math]}$ can be calculated with the STDMED function in a DATA step)

$\text{[math]}$

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

$\text{[math]}$

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 of the standard normal distribution

$\text{[math]}$

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

Plotted Points

Each point on a median chart indicates the value of a subgroup median ( $\text{[math]}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is $\text{[math]}$ . Each point on a range chart indicates the value of a subgroup range ( $\text{[math]}$ ). For example, the value plotted for the tenth subgroup is $\text{[math]}$ .

Central Lines

On a median chart, the value of the central line indicates an estimate for $\text{[math]}$ , which is computed as

$\text{[math]}$ by default
$\text{[math]}$ when you specify MEDCENTRAL=AVGMEAN
$\text{[math]}$ when you specify MEDCENTRAL=MEDMED
$\text{[math]}$ when you specify $\text{[math]}$ with the MU0= option

On the range chart, 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 $\text{[math]}$ . The central line on the range chart varies with $\text{[math]}$ .

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 its limits

The following table provides the formulas for the limits:

Table 15.32 Limits for Median and Range Charts
Control Limits
Median Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$
Range Chart	LCL $\text{[math]}$ lower control limit = $\text{[math]}$
	UCL $\text{[math]}$ upper control limit = $\text{[math]}$
Probability Limits
Median Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$
Range Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$

In Table 15.32, replace $\text{[math]}$ with $\text{[math]}$ if you specify MEDCENTRAL=AVGMEAN, and replace $\text{[math]}$ with $\text{[math]}$ if you specify MEDCENTRAL=MEDMED. Replace $\text{[math]}$ with $\text{[math]}$ if you specify $\text{[math]}$ with the MU0= option, and replace $\text{[math]}$ with $\text{[math]}$ if you specify $\text{[math]}$ with the SIGMA0= option.

The formulas assume that the data are normally distributed. Note that the limits for both charts vary with $\text{[math]}$ and that the probability limits for $\text{[math]}$ 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 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.

MRCHART Statement: SHEWHART Procedure

Constructing Charts for Medians and Ranges

Plotted Points

Central Lines

Control Limits