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August 31, 2006
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114 Computing Process Capability Indices for Nonnormal Data
nonconforming items to derive the corresponding value of C p . The bias and dispersion
of the estimated C p (u,v) values are compared with the target C p values. Rivera et al. 20
varied the upper specification limits of the underlying distributions to derive the
actual number of nonconforming items and the equivalent C pk values. Estimated
C pk values calculated from the transformed simulation data are compared with these
target C pk values.
In practice, PCIs are commonly used for tracking performance and comparison
between different processes. Although such uses without examining the underlying
distribution should not be encouraged, a “good” surrogate PCI for nonnormal data
should be compatible with the PCI computed under normality when the correspond-
ing fractions nonconforming are about the same. This motivates a scheme similar
to that of Rivera et al. 20 where the fraction nonconforming is fixed a priori by using
suitable specification limits, and PCIs computed using various methods are then com-
pared with a target value. This leads to consideration of unilateral tolerance, where
C pu , a single tolerance limit capability index, is used as the comparison yardstick in
our simulation study. For a targeted C pu value the fraction of nonconforming units
under the normality assumption can be determined using

fraction nonconforming = (−3C pu ) . (9.13)
In our simulation study, targeted values of C pu = 1, 1.5 and 1.667 are used and the
corresponding USL values for lognormal and Weibull distributions with the same frac-
tion nonconforming are obtained. These USL values are then used to estimate the C pu
index pertaining to the different methods from the simulated data. These estimated
C pu s are then compared with the targeted C pu values.
A superior method is one with its sample mean of the estimated C pu having the
smallest deviation from the target value (accuracy) and with the smallest variability,
measured by the spread or standard deviation of the estimated C pu values (precision).
A graphical representation that conveniently depicts these two characteristics is the
simple box-and-whisker plot.

9.3.2 Underlying distributions
Lognormal and Weibull distributions are used to investigate the effect of nonnormal
data on the PCIs, as they are known to have parameter values that can represent slight,
moderate, and severe departures from normality. These distributions are also known
to have significantly different tail behaviors, which greatly influences the process
capability. The sets of parameter values used in the simulation study are given in
Table 9.1. To save space, detailed results are presented here only for those sets in bold
type. The skewness and kurtosis for each of these distributions are given in Table 9.2.
The cumulative distribution functions (CDFs) of the lognormal and Weibull distri-
butions are respectively given by

ln x − μ
F (x; μ, σ) = , x ≥ 0, (9.14)
σ
x
n

F (x; η, σ) = 1 − exp − , x ≥ 0. (9.15)
σ