Distributional properties of estimated capability indices based on subsamples

Abstract

AbstractUnder the assumption of normality, the distribution of estimators of a class of capability indices, containing the indices $C_p$, $C_{pk}$, $C_{pm}$ and $C_{pmk}$, is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the process variance is estimated using the pooled variance from subsamples collected over time for an in‐control process. The derived theory is then applied to study the use of hypothesis testing to assess process capability. Numerical investigations are made to explore the effect of the size and number of subsamples on the efficiency of the hypothesis test for some indices in the studied class. The results for $C_{pm}$ and $C_{pk}$ indicate that, even when the total number of sampled observations remains constant, the power of the test decreases as the subsample size decreases. It is shown how the power of the test is dependent not only on the subsample size and the number of subsamples, but also on the relative location of the process mean from the target value. As part of this investigation, a simple form of the cumulative distribution function for the non‐central $\chi^2$‐distribution is also provided. Copyright © 2003 John Wiley & Sons, Ltd.