1. Very Low Defect Occurrence Rates
When the quality of a process is improved to the point where it approaches very low defect rates standard p, c and u charts will no longer provide meaningful information about the stability of the process.
When defects are so rare that the Upper Control Limit is less than 1 (c < .09) any occurrence will constitute an ‘out-of-control’ condition…obviously a different type of chart is needed.
When occurrence rates are low, the sequential number of opportunities for a defect that occur without a defect will have a (roughly) Geometric Distribution.
A control chart can be constructed using this distribution:
The number of zero defect opportunities that occur between defects is counted = a RUN.
Opportunities can be individual units, man-hours, operating hours, production volumes, etc.
The formulas are:
An out-of-control condition exists when:
1 point falls below the LCL (above the limit is an improved condition)
5 points fall consecutively below the Center Line (below the center line is an improved condition)
a is the significance level
Beverly Daniels

2. Very Low Defect Rate Example
CONTROL
Very Low Defect Rate Example
A high quality process produces 1558 units with only 10 defects.
A traditional c chart would result in the following limits:
If the table at the right lists the number of units produced until a defect is found, it is obvious that a c chart will be a silly looking chart that will provide no useful data.
If we apply the geometric distribution and select an a of 5%, we get the following limits:
The resulting Control Chart shows a stable process:
Beverly Daniels