Constructing Charts for Proportion Nonconforming (p Charts)

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Constructing Charts for Proportion Nonconforming (p Charts)

The following notation is used in this section:

expected proportion of nonconforming items produced by the process

$p_{i}$

proportion of nonconforming items in the ith subgroup

$X_{i}$

number of nonconforming items in the ith subgroup

$n_{i}$

number of items in the ith subgroup

$\bar{p}$

average proportion of nonconforming items taken across subgroups:

$\bar{p} = \frac{n_1p_1 + \cdots + n_ Np_ N}{n_1 + \cdots + n_ N} = \frac{X_1 + \cdots + X_ N}{n_1 + \cdots + n_ N}$

number of subgroups

$I_{T}(\alpha ,\beta )$

incomplete beta function:

$I_{T}(\alpha ,\beta ) = (\Gamma (\alpha +\beta )/\Gamma (\alpha )\Gamma (\beta )) \int _{0}^{T}t^{\alpha - 1}(1-t)^{\beta -1}dt$

for , $\alpha >0$ , and $\beta >0$ , where $\Gamma (\cdot )$ is the gamma function

Plotted Points

Each point on a p chart represents the observed proportion () of nonconforming items in a subgroup. For example, suppose the second subgroup (see Figure 17.68) contains 16 items, of which two are nonconforming. The point plotted for the second subgroup is $p_{2} = 2/16=0.125$ .

Figure 17.68: Proportions Versus Counts

Note that an chart displays the number (count) of nonconforming items $X_{i}$ . You can use the NPCHART statement to create charts; see NPCHART Statement: SHEWHART Procedure.

Central Line

By default, the central line on a p chart indicates an estimate of p that is computed as $\bar{p}$ . If you specify a known value ( $p_{0}$ ) for p, the central line indicates the value of $p_{0}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of above and below the central line. The default limits are computed with k = 3 (these are referred to as $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that $p_{i}$ exceeds the limits

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

$\displaystyle \mbox{LCL}$	$\displaystyle =$	$\displaystyle \mbox{max} \left(\bar{p} - k\sqrt {\bar{p}(1-\bar{p})/n_ i}\; , 0 \right)$
$\displaystyle \mbox{UCL}$	$\displaystyle =$	$\displaystyle \mbox{min}\left(\bar{p} + k\sqrt {\bar{p}(1-\bar{p})/n_ i}\; , 1 \right)$

A lower probability limit for can be determined using the fact that

$\begin{array}{ll} P\{ p_ i < \mbox{LCL}\} & = 1 - P\{ p_ i \geq \mbox{LCL}\} \\ & = 1 - P\{ X_ i \geq n_ i\mbox{LCL}\} \\ & = 1 - I_{\bar{p}}(n_ i\mbox{LCL},n_ i+1-n_ i\mbox{LCL}) \\ & = I_{1- \bar{p}}(n_ i+1-n_ i\mbox{LCL},n_ i\mbox{LCL}) \\ \end{array}$

Refer to Johnson, Kotz, and Kemp (1992). This assumes that the process is in statistical control and that is binomially distributed. The lower probability limit LCL is then calculated by setting

$I_{1- \bar{p}}(n_ i+1-n_ i\mbox{LCL},n_ i\mbox{LCL}) = \alpha /2$

and solving for LCL. Similarly, the upper probability limit for can be determined using the fact that

$\begin{array}{ll} P\{ p_ i > \mbox{UCL}\} & = P\{ p_ i > \mbox{UCL}\} \\ & = P\{ X_ i > n_ i\mbox{UCL}\} \\ & = I_{\bar{p}}(n_ i\mbox{UCL},n_ i+1-n_ i\mbox{UCL}) \\ \end{array}$

The upper probability limit UCL is then calculated by setting

$I_{\bar{p}}(n_ i\mbox{UCL},n_ i+1-n_ i\mbox{UCL}) = \alpha /2$

and solving for UCL. The probability limits are asymmetric around the central line. Note that both the control limits and probability limits vary with $n_{i}$ .

You can specify parameters for the limits as follows:

Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
Specify $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $p_{0}$ with the P0= option or with the variable _P_ in a LIMITS= data set.