Constructing Charts for Proportion Nonconforming (p Charts)

The following notation is used in this section:

p	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: $\bar{p} = \frac{n_1p_1 + \cdots + n_ Np_ N}{n_1 + \cdots + n_ N} = \frac{X_1 + \cdots + X_ N}{n_1 + \cdots + n_ N}$
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 $0<T<1$ , $\alpha >0$ , and $\beta >0$ , where $\Gamma (\cdot )$ is the gamma function

Plotted Points

Each point on a p chart represents the observed proportion ( $p_ i=X_ i/n_ i$ ) 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 $np$ chart displays the number (count) of nonconforming items $X_{i}$ . You can use the NPCHART statement to create $np$ 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 $p_ i$ 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

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

A lower probability limit for $p_ i$ 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 $X_ i$ 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 $p_ i$ 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.