PROC SHEWHART: Constructing Charts for Proportion Nonconforming (p Charts)

The SHEWHART Procedure

Constructing Charts for Proportion Nonconforming (p Charts)

The following notation is used in this section:

$\text{[math]}$

expected proportion of nonconforming items produced by the process

$\text{[math]}$

proportion of nonconforming items in the $\text{[math]}$ th subgroup

$\text{[math]}$

number of nonconforming items in the $\text{[math]}$ th subgroup

$\text{[math]}$

number of items in the $\text{[math]}$ th subgroup

$\text{[math]}$

average proportion of nonconforming items taken across subgroups:

$\text{[math]}$

number of subgroups

$\text{[math]}$

incomplete beta function:

$\text{[math]}$

for $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ , where $\text{[math]}$ is the gamma function

Plotted Points

Each point on a $\text{[math]}$ chart represents the observed proportion ( $\text{[math]}$ ) of nonconforming items in a subgroup. For example, suppose the second subgroup (see Figure 13.21.12) contains 16 items, of which two are nonconforming. The point plotted for the second subgroup is $\text{[math]}$ .

Output 13.21.12 Proportions Versus Counts

Note that an $\text{[math]}$ chart displays the number (count) of nonconforming items $\text{[math]}$ . You can use the NPCHART statement to create $\text{[math]}$ charts; see NPCHART Statement

Central Line

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

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 lower and upper control limits, LCL and UCL, respectively, are computed as

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

A lower probability limit for $\text{[math]}$ can be determined using the fact that

$\text{[math]}$

Refer to Johnson, Kotz, and Kemp (1992). This assumes that the process is in statistical control and that $\text{[math]}$ is binomially distributed. The lower probability limit LCL is then calculated by setting

$\text{[math]}$

and solving for LCL. Similarly, the upper probability limit for $\text{[math]}$ can be determined using the fact that

$\text{[math]}$

The upper probability limit UCL is then calculated by setting

$\text{[math]}$

and solving for UCL. The probability limits are asymmetric around the central line. Note that both the control limits and probability limits vary with $\text{[math]}$ .

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 P0= option or with the variable _P_ in a LIMITS= data set.

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