PROC SHEWHART: Constructing Box Charts

The SHEWHART Procedure

Constructing Box Charts

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

$\text{[math]}$

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:

$\text{[math]}$

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)

$\text{[math]}$

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

$\text{[math]}$

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

$\text{[math]}$

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

Elements of Box-and-Whisker Plots

A box-and-whisker plot is displayed for the measurements in each subgroup on the box chart. Figure 13.14 illustrates the elements of each plot.

Figure 13.14 Box-and-Whisker Plot

The skeletal style of the box-and-whisker plot shown in Figure 13.14 is the default. You can specify alternative styles with the BOXSTYLE= option; see Example 13.2 or the entry for BOXSTYLE= in Dictionary of Options

Control Limits and Central Line

You can compute the limits in the following ways:

as a specified multiple ( $\text{[math]}$ ) of the standard error of $\text{[math]}$ (or $\text{[math]}$ ) 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 CONTROLSTAT= option specifies whether control limits are computed for subgroup means (the default) or subgroup medians. The following tables provide the formulas for the limits:

Table 13.5 Control Limits and Central Line for Box Charts
CONTROLSTAT=MEAN	CONTROLSTAT=MEDIAN
LCLX $\text{[math]}$ lower limit = $\text{[math]}$	LCLM $\text{[math]}$ lower limit = $\text{[math]}$
Central Line = $\text{[math]}$	Central Line = $\text{[math]}$
UCLX $\text{[math]}$ upper limit = $\text{[math]}$	UCLM $\text{[math]}$ upper limit = $\text{[math]}$

Table 13.6 Probability Limits and Central Line for Box Charts
CONTROLSTAT=MEAN	CONTROLSTAT=MEDIAN
LCLX $\text{[math]}$ lower limit = $\text{[math]}$	LCLM $\text{[math]}$ lower limit = $\text{[math]}$
Central Line = $\text{[math]}$	Central Line = $\text{[math]}$
UCLX $\text{[math]}$ upper limit = $\text{[math]}$	UCLM $\text{[math]}$ upper limit = $\text{[math]}$

In the preceding tables, replace $\text{[math]}$ with $\text{[math]}$ if you specify MEDCENTRAL=AVGMEAN in addition to CONTROLSTAT=MEDIAN. Likewise, replace $\text{[math]}$ with $\text{[math]}$ if you specify MEDCENTRAL=MEDMED in addition to CONTROLSTAT=MEDIAN. 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 13.5 and Table 13.6.

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

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

Note:You can suppress the display of the control limits with the NOLIMITS option. This is useful for creating standard side-by-side box-and-whisker plots.

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