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Calculating confidence intervals for a quality figure of merit

T. E. Daughters
Journal / Anthology

Mathematics with Vision: Proceedings of the First International Mathematica Symposium
Year: 1995
Page range: 119-126

Quality initiatives at AT&T as well as other companies include the measurement of a statistical figure of merit Cpk. This figure of merit indicates the number of standard deviations between a product's typical or mean specification and the minimum or maximum limit. The higher Cpk is, the more tightly the product specification is clustered about its mean, and the less likely that performance will be observed near its limits. Because measuring Cpk involves working with random data, there is always uncertainty in the value calculated. It is useful to quantify the amount of uncertainty in a particular Cpk measurement. Calculating confidence intervals for means and variances based on sampled data is well known and directly available from one of the standard Mathematica packages. Calculating a confidence interval for Cpk is not so straight forward, because the figure or merit involves both the mean and standard deviation as well as a constant (the design limit). With some manipulation, measurements of Cpk based on sampled data can be shown to follow the non-central student-t distribution between the observed value and infinity. The author has performed this calculation for many sample sizes and a wide range of useful Cpk values. The results are presented in a unique graphical format where the level of confidence is read directly from a graph given the measure value of Cpk and the sample size. In addition, an approximate method for finding this confidence interval is presented which is accurate for sample sizes greater than 25. This method is iterative, but the iterations can be easily programmed into a Mathematica function.

*Mathematics > Probability and Statistics