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 CCHART Statement

## Constructing Charts for Numbers of Nonconformities (c Charts)

The following notation is used in this section:
 u expected number of nonconformities per unit produced by the process ui number of nonconformities per unit in the i th subgroup ci total number of nonconformities in the i th subgroup ni number of inspection units in the i th subgroup. Typically, ni = 1 and ui=ci for c charts. In general, ui=ci/ni. average number of nonconformities per unit taken across subgroups. The quantity is computed as a weighted average: N number of subgroups has a central distribution with degrees of freedom

### Plotted Points

Each point on a c chart represents the total number of nonconformities (ci) in a subgroup. For example, Figure 33.10 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 i th subgroup is denoted by ni, which is the subgroup sample size. The value of ni can be fractional; Figure 33.10 shows n3=2.5 units in the third subgroup.

Figure 33.10: Terminology for c Charts and u Charts

The number of nonconformities in the i th subgroup is denoted by ci. The number of nonconformities per unit in the i th subgroup is denoted by ui=ci/ni. In Figure 33.10, the number of welds per inspection unit in the third subgroup is u3=2/2.5=0.8.

A u chart created with the UCHART statement plots the quantity ui for the i th subgroup (see Chapter 41). An advantage of a u chart is that the value of the central line at the i th subgroup does not depend on ni. This is not the case for a c chart, and consequently, a u chart is often preferred when the number of units ni is not constant across subgroups.

### Central Line

On a c chart, the central line indicates an estimate for niu, which is computed as .If you specify a known value (u0) for u, the central line indicates the value of niu0.

Note that the central line varies with subgroup sample size ni. When ni=1 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 (k) of the standard error of ci 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 ci 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 ni. If ni=1 for all subgroups, the control limits have constant values.
An upper probability limit UCLC for ci 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 ci 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 (ni) and are asymmetric about the central line.

If a standard value u0 is available for u, replace with u0 in the formulas for the control limits. 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 u0 with the U0= option or with the variable _U_ in a LIMITS= data set.

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