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August 31, 2006
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JWBK119-09
128 Computing Process Capability Indices for Nonnormal Data
√
Table 9.7 Practitioners’ guide: methods applicable for each defined range of skewness β 1
and kurtosis (β 2 ).
√ √ √
−0.1 < β 1 ≤ 0.1 0.1 < β 1 ≤ 0.1 β 1 > 0.1
2.5 <β 2 ≤ 3.5 AN CM PP BC BC
PP BC DF JT JT
DF JT WI
WV WI
β 2 > 3.5 BC BC BC
JT JT JT

PP, probability plot; AN, assumed normal; DF, distribution-free tolerance interval;
WV, weighted variance; WI, Wright’s index; CM, Clements’ method (n ≥ 100);
BC, Box--Cox transformation (n ≥ 100); JT, Johnson transformation (n ≥ 100).

2. For the purpose of capability analysis, the success of a transformation method
depends on its ability to capture the tail behavior of the distribution (beyond the
USL percentile point), particularly for high-yield or highly capable processes (see
case studies in Chapter 10).

This means that to apply transformation methods in a process capability study, we
should have n ≥ 100.
Nevertheless, the superiority of the Box--Cox transformation is readily seen from the
box plots, especially for the lognormal distribution (see Figures 9.3--9.8). This is to be
expected, since the Box--Cox transformation method provides an exact transformation
by taking the natural logarithms of the lognormal variates to give a normal distribu-
tion. Comparing the Box--Cox and Johnson transformation methods, the Box--Cox
is more accurate and precise. However, both methods are computer-intensive, with
the Box--Cox method requiring maximization of a log-likelihood function to obtain
an optimal λ value and the Johnson method requiring fitting of the first four moments
to determine the appropriate Johnson family. This leads us to suggest that the Box--
Cox transformation is the preferred method for handling nonnormal data whenever
a computer-assisted analysis is available.
Combining these results, we present in Table 9.7 a practitioners’guide which shows
the methods applicable for each defined range of skewness and kurtosis (see Table
9.2). Note that these ranges are not exact; rather they are inferred from the simulation
results available. In Table 9.7, methods that are considered “superior”, namely those
that give more accurate and precise estimates of C pu , are shown in bold type. Methods
that are not highlighted in this way are recommended on the basis of their case of
calculation and small bias in estimating the C pu index. It is evident that the Box--
Cox transformation is consistently superior in performance throughout. The use of
probability plotting is recommended for processes that exhibit a mild departure from
normality. Furthermore, a probability plot allows the assessment of the goodness of
fit of a particular probability model to a set of data.

9.5 CONCLUSION

Seven methods for computing PCIs have been reviewed and their performance eval-
uated by Monte Carlo simulation. In general, methods involving transformation,