Properties and performance of thec-chart for attributes data

S. Chakraborti · Schalk William Human

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Advanced Statistical Process Monitoring Probability and Statistical Research Fault Detection and Control Systems

Abstract

The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL0=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.

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References

1990Statistical Quality Control 1993The Effect of Sample Size on Estimated Limits forXandXControl Charts 1997Control Charts Based on Attribute Data: Bibliography and Review 2003Statistical Quality Control 2000Run length, average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning 1981Shewhart x-charts with estimated process variance 2006Parameter Estimation and Performance of the $p$ -Chart for Attributes Data

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