Constructing Charts for Number Nonconforming (np Charts)

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Constructing Charts for Number Nonconforming (np 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 an chart represents the observed number () of nonconforming items in a subgroup. For example, suppose the first subgroup (see Figure 17.58) contains 12 items, of which three are nonconforming. The point plotted for the first subgroup is $X_{1} = 3$ .

Figure 17.58: Proportions Versus Counts

Note that a p chart displays the proportion of nonconforming items $p_{i}$ . You can use the PCHART statement to create p charts; see PCHART Statement: SHEWHART Procedure.

Central Line

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

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of $X_{i}$ 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 $X_{i}$ exceeds the limits

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

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

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

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

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

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

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

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

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

The upper probability limit UCL is then calculated by setting

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

and solving for UCL. The probability limits are asymmetric about 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 the LIMITS= data set.

NPCHART Statement: SHEWHART Procedure

Constructing Charts for Number Nonconforming (np Charts)

Plotted Points

Central Line

Control Limits