CCHART Statement: SHEWHART Procedure
Constructing Charts for Numbers of Nonconformities (c Charts)
The following notation is used in this section:
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expected number of nonconformities per unit produced by the process |
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number of nonconformities per unit in the |
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total number of nonconformities in the |
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number of inspection units in the |
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average number of nonconformities per unit taken across subgroups. The quantity
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number of subgroups |
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has a central |
and 
for 
distribution with 
degrees of freedom Plotted Points
Each point on a 
chart represents the total number of nonconformities (
) in a subgroup. For example, Figure 15.24 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 15.24 shows 
units in the third subgroup.
Charts and 
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 15.24, 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: SHEWHART Procedure). 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
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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.
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An upper probability limit UCLC for can be determined using the fact that
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The upper probability limit UCLC is then calculated by setting
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and solving for UCLC.
A similar approach is used to calculate the lower probability limit LCLC, using the fact that
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The lower probability limit LCLC is then calculated by setting
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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.