Constructing Charts for Proportion Nonconforming (p Charts)

PCHART Statement

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 i^th subgroup

X_i number of nonconforming items in the i^th subgroup

n_i number of items in the i^th subgroup

$\bar{p}$ average proportion of nonconforming items taken across subgroups:
$\bar{p} = \frac{n_1p_1 + ... + n_Np_N} {n_1 + ... + n_N} = \frac{X_1 + ... + X_N} {n_1 + ... + n_N}$

N number of subgroups

$I_{T}(\alpha,\beta)$ incomplete beta function:
$I_{T}(\alpha,\beta) = (\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)) \int_{0}^Tt^{\alpha - 1}(1-t)^{\beta-1}dt$
for 0<T<1, $\alpha\gt$ , and $\beta\gt$ , 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 38.10) contains 16 items, of which two are nonconforming. The point plotted for the second subgroup is p₂ = 2/16=0.125.

Figure 38.10: 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 Chapter 37, "NPCHART Statement."

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 (p₀) for p, the central line indicates the value of p₀.

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

${LCL} &= &{max} (\bar{p} - k\sqrt{\bar{p}(1-\bar{p})/n_i}\;, 0 ) \ {UCL} &= &{min}(\bar{p} + k\sqrt{\bar{p}(1-\bar{p})/n_i}\;, 1 )$

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

$P\{p_i \lt {LCL}\} & = 1 - P\{p_i \geq {LCL}\} \ & = 1 - P\{X_i \geq n_i{LCL}\} ... ...bar{p}}(n_i{LCL},n_i+1-n_i{LCL}) \ & = I_{1- \bar{p}}(n_i+1-n_i{LCL},n_i{LCL}) \$

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

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

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

$P\{p_i \gt {UCL}\} & = P\{p_i \gt {UCL}\} \ & = P\{X_i \gt n_i{UCL}\} \ & = I_{\bar{p}}(n_i{UCL},n_i+1-n_i{UCL}) \$

The upper probability limit UCL is then calculated by setting

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

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 p₀ with the P0= option or with the variable _P_ in a LIMITS= data set.

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p	expected proportion of nonconforming items produced by the process
p_i	proportion of nonconforming items in the i^th subgroup
X_i	number of nonconforming items in the i^th subgroup
n_i	number of items in the i^th subgroup
$\bar{p}$	average proportion of nonconforming items taken across subgroups: $\bar{p} = \frac{n_1p_1 + ... + n_Np_N} {n_1 + ... + n_N} = \frac{X_1 + ... + X_N} {n_1 + ... + n_N}$
N	number of subgroups
$I_{T}(\alpha,\beta)$	incomplete beta function: $I_{T}(\alpha,\beta) = (\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)) \int_{0}^Tt^{\alpha - 1}(1-t)^{\beta-1}dt$ for 0<T<1, $\alpha\gt$ , and $\beta\gt$ , where $\Gamma(\cdot)$ is the gamma function