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The SHEWHART Procedure

Constructing Charts for Numbers of Nonconformities (c Charts)

The following notation is used in this section:

expected number of nonconformities per unit produced by the process

number of nonconformities per unit in the th subgroup

total number of nonconformities in the th subgroup

number of inspection units in the th subgroup. Typically, and for charts. In general, .

average number of nonconformities per unit taken across subgroups. The quantity is computed as a weighted average:

     

number of subgroups

has a central distribution with degrees of freedom

Plotted Points

Each point on a chart represents the total number of nonconformities () in a subgroup. For example, Figure 13.7.13 displays three sections of pipeline that are inspected for defective welds (indicated by an X). Each section represents a subgroup composed of a number of inspection units, which are 1000-foot-long sections. The number of units in the th subgroup is denoted by , which is the subgroup sample size. The value of can be fractional; Figure 13.7.13 shows units in the third subgroup.

Output 13.7.13 Terminology for Charts and Charts
Terminology for c Charts and u Charts

The number of nonconformities in the th subgroup is denoted by . The number of nonconformities per unit in the th subgroup is denoted by . In Figure 13.7.13, the number of welds per inspection unit in the third subgroup is .

A chart created with the UCHART statement plots the quantity for the th subgroup (see UCHART Statement). An advantage of a chart is that the value of the central line at the th subgroup does not depend on . This is not the case for a chart, and consequently, a chart is often preferred when the number of units is not constant across subgroups.

Central Line

On a chart, the central line indicates an estimate for , which is computed as . If you specify a known value () for , the central line indicates the value of .

Note that the central line varies with subgroup sample size . When for all subgroups, the central line has the constant value .

Control Limits

You can compute the limits in the following ways:

  • as a specified multiple () of the standard error of above and below the central line. The default limits are computed with (these are referred to as limits).

  • as probability limits defined in terms of , a specified probability that exceeds the limits

The lower and upper control limits, LCLC and UCLC respectively, are given by

     
     

The upper and lower control limits vary with the number of inspection units per subgroup . If for all subgroups, the control limits have constant values.

     
     

An upper probability limit UCLC for can be determined using the fact that

     

The upper probability limit UCLC is then calculated by setting

     

and solving for UCLC.

A similar approach is used to calculate the lower probability limit LCLC, using the fact that

     

The lower probability limit LCLC is then calculated by setting

     

and solving for LCLC. This assumes that the process is in statistical control and that has a Poisson distribution. For more information, refer to Johnson, Kotz, and Kemp (1992). Note that the probability limits vary with the number of inspection units per subgroup () and are asymmetric about the central line.

If a standard value is available for , replace with in the formulas for the control limits. You can specify parameters for the limits as follows:

  • Specify 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 with the U0= option or with the variable _U_ in a LIMITS= data set.

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