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13

A Graphical Approach to

Obtaining Confidence

Limits of C pk

L. C. Tang, S. E. Than and B. W. Ang

The process capability index C pk has been widely used as a process performance
measure. In practice this index is estimated using sample data. Hence it is of great
interest to obtain confidence limits for the actual index given a sample estimate. In this
chapter we depict graphically the relationship between the process potential index
(C p ), the process shift index (k) and the percentage nonconforming (p). Based on the
monotone properties of the relationship, we derive two-sided confidence limits for
kand C pk under two different scenarios. These two limits are combined using the
Bonferroni inequality to generate a third type of confidence limit. The performance
of these limits of C pk in terms of their coverage probability and average width is
evaluated by simulation. The most suitable type of confidence limit for each specific
range of k is then determined. The usage of these confidence limits is illustrated
with examples. Finally, a performance comparison is done between the proposed
confidence limits and three non-parametric bootstrap confidence limits. The results
show that the proposed method consistently gives the smallest width and yet provides
the intended coverage probability.

This chapter is based on the article by L. C. Tang, S. E. Than and B. W. Ang, ‘A graphical approach to
obtaining confidence limits of C pk ’, Quality and Reliability Engineering International, 13(6), 1997, pp. 337--346,
and is reproduced by the permission of the publisher, John Wiley & Sons, Ltd
Six Sigma: Advanced Tools for Black Belts and Master Black Belts L. C. Tang, T. N. Goh, H. S. Yam and T. Yoap
C 2006 John Wiley & Sons, Ltd
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