An Approximation Approach for Making Decisions in Assessing the Capability Index C_pk from the Subsamples

ABSTRACT

Process capability indices are widely used to measure the ability of the process to manufacture products meeting established specification limits. Simply reporting and then making a decision from the calculated estimate of the capability index along is not reliable since sampling errors are ignored. Under the assumption of normality, the test for making decisions in assessing C _pk index based on a newly proposed estimator is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the pooled standard deviation (multiplied by appropriate unbiasing factors) is substituted for the process standard deviation. The method to derive an associated operating characteristic curve is examined. It is demonstrated that the procedure in obtaining an estimator of C _pk can be adequately simplified which bears an approximate normal distribution and hence the more complicated use of the non central t-distribution can be avoided. Moreover, applying two-point adjustment, an equation to determine the critical value and estimate the sample size necessary to achieve the recommended minimal value for a C _pk index is provided.

Key Words:

• C _pk index
• Hamaker's approximation to the noncentral t-distribution
• Noncentral t-distribution
• Process mean
• Process standard deviation
• The power of the test

Acknowledgments

The author would like to thank the referees for helpful comments and suggestions which significantly improved this article. This study is supported in part by the National Science Council, Taiwan, R.O.C. (Grant No. NSC 92-2213-E-238-014).