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 NPCHART Statement

## Constructing Charts for Number Nonconforming (np Charts)

The following notation is used in this section:
 p expected proportion of nonconforming items produced by the process pi proportion of nonconforming items in the i th subgroup Xi number of nonconforming items in the i th subgroup ni number of items in the i th subgroup average proportion of nonconforming items taken across subgroups: N number of subgroups incomplete beta function: for 0

### 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.

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 .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:

• as a specified multiple (k) of the standard error of Xi above and below the central line. The default limits are computed with k=3 (these are referred to as limits).
• as probability limits defined in terms of , a specified probability that Xi exceeds the limits

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

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

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

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

The upper probability limit UCL is then calculated by setting

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:
• Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
• Specify with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
• Specify a constant nominal sample size 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 the LIMITS= data set.

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