00394 �4bf53b90d85a9db56d6772e2aee78a

398

Obenchain

Table 4. Yields for Processes with Mean on Target and Target at the Tolerance Mid-Point

................Cp..............	Normal Theory Yield	Distribution-Free Minimum Yield
0.666	95.45%	75.00%
1.000	99.73%	88.89%
1.333	99.994%	93.75%
1.666	99.999?%	96.00%
2.000	99.9999?%	97.22%

single process over time, Cp-type statistics can lack cogency in comparisons across processes.

In stark contrast, cumulative capabilities do not depend on the appropriateness of process specification limits; can be based on any functional form of regret; do measure process yields for an entire class of intervals; and do express results on a natural, cost-unitless scalę applicable across (as well as within) processes. In addition to CC confidence limits on the marginal distribution of regret indices, as described in [16], highest posterior density intervals for time-ordered regret indices (even when composite equivalent expectancies vary with reporting period) are provided by QMP, Hoadley (1981,1986).

For readers who can't resist the temptation to pick a single value off of the cumulative capability curve as a capability summary statistic, I recommend the ordinate at I = 1:

CC( 1) = Prob[ random regret index < 1 ]    [27]

CC( 1 ) is the expected proportion of measurements taken on the process that will correspond to either expected or better performance. Although regret indices as deflned here always have expected value (approximately) one, the probability of observing an index value less than one will be considerably greater than 0.5 when the process has "high" capability.

Figurę 15 and Table 5, below, give theoretical values of CC( 1 ) as a function of EE for Poisson and standardized gamm a distributions. Calculations were performed using the PROBGAM() function of SAS (1990); specifically, the incomplete gamma function ratio is: