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NPCHART Statement

Constructing Charts for Number Nonconforming (np Charts)

The following notation is used in this section:
pexpected proportion of nonconforming items produced by the process
piproportion of nonconforming items in the i th subgroup
Xinumber of nonconforming items in the i th subgroup
ninumber 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}
Nnumber 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 an np chart represents the observed number (Xi) of nonconforming items in a subgroup. For example, suppose the first subgroup (see Figure 37.9) contains 12 items, of which three are nonconforming. The point plotted for the first subgroup is X1 = 3.

balls.gif (4650 bytes)

Figure 37.9: Proportions Versus Counts

Note that a p chart displays the proportion of nonconforming items pi. You can use the PCHART statement to create p charts; see Chapter 38, "PCHART Statement."

Central Line

By default, the central line on an np chart indicates an estimate for nip, which is computed as n_i\bar{p}.If you specify a known value (p0) for p, the central line indicates the value of nip0. Note that the central line varies with ni.

Control Limits

You can compute the limits in the following ways:


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

{LCL} = {max}(n_{i}\bar{p} -
 k\sqrt{n_{i}\bar{p}(1-\bar{p})}\;, 0 ) \ {UCL} = {min}(n_{i}\bar{p} +
 k\sqrt{n_{i}\bar{p}(1-\bar{p})}\;, n_{i} )

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

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

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

I_{1- \bar{p}}(n_i+1-{LCL},{LCL}) = \alpha/2
and solving for LCL. Similarly, the upper probability limit for Xi can be determined using the fact that
P\{X_i \gt {UCL}\} & = P\{X_i \gt {UCL}\} \ & = I_{\bar{p}}({UCL},n_i+1-{UCL}) \

The upper probability limit UCL is then calculated by setting

I_{\bar{p}}({UCL},n_i+1-{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 ni. You can specify parameters for the limits as follows:

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