PCHART Statement: SHEWHART Procedure

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.