PROC SHEWHART: Constructing Charts for Means and Standard Deviations

The SHEWHART Procedure

Constructing Charts for Means and Standard Deviations

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

$\text{[math]}$

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

$\text{[math]}$

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

$\text{[math]}$

number of subgroups

$\text{[math]}$

weighted average of subgroup means

$\text{[math]}$

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

$\text{[math]}$

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 mean ( $\text{[math]}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 13, the mean plotted for this subgroup is

$\text{[math]}$

Each point on an $\text{[math]}$ chart indicates the value of a subgroup standard deviation ( $\text{[math]}$ ). For example, the standard deviation plotted for the tenth subgroup is

$\text{[math]}$

Central Lines

On an $\text{[math]}$ chart, by default, the central line indicates an estimate of $\text{[math]}$ , which is computed as

$\text{[math]}$

If you specify a known value ( $\text{[math]}$ ) for $\text{[math]}$ , the central line indicates the value of $\text{[math]}$ .

On the $\text{[math]}$ 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 $\text{[math]}$ . 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 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 13.77 Limits for $\text{[math]}$ and $\text{[math]}$ Charts
Control Limits
$\text{[math]}$ Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$
$\text{[math]}$ Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$

Probability Limits
$\text{[math]}$ Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$
$\text{[math]}$ Chart	LCL $\text{[math]}$ lower limit = $\text{[math]}$
	UCL $\text{[math]}$ upper limit = $\text{[math]}$

The formulas for $\text{[math]}$ charts assume that the data are normally distributed. If standard values $\text{[math]}$ and $\text{[math]}$ are available for $\text{[math]}$ and $\text{[math]}$ , respectively, replace $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$ with $\text{[math]}$ in Table 13.77. Note that the limits vary with $\text{[math]}$ and that the probability limits for $\text{[math]}$ are asymmetric about 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 a LIMITS= data set.
Specify $\text{[math]}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

Top of Page