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August 31, 2006
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JWBK119-09
110 Computing Process Capability Indices for Nonnormal Data
where w is the width of the tolerance interval with 99.73% coverage 95% of the time,
w 2 is the width of the tolerance interval with 95.46% coverage 95% of the time, and w 3
is the width of the tolerance interval with 68.26% coverage 95% of the time. The order
statistics estimates of w, w 2 and w 3 , based on the normality assumption, are given
6
by Chan et al. This is a more conservative method, since the natural process width
is greater as it is estimated taking the sampling variation into account. This is the
only method considered here that will give a different result even if the underlying
distributionisnormal.Itisincludedheretoinvestigatewhetheritsconservativenature
is preserved under nonnormality.

9.2.3 Weighted variance method
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Choi and Bai proposed a heuristic weighted variance method to adjust the PCI values
according to the degree of skewness of the underlying population. Let P x be the
probability that the process variable X is less than or equal to its mean μ,
n
1
¯
P x = I(X − X i ),
n
i=1
where I(x) = 1if x > 0 and I(x) = 0if x < 0. The PCI based on the weighted variance
method is defined as
USL − LSL
C p =
6σ W x
√
where W x = 1 +|1 − 2P x |. Also
USL − μ
C pu = √ ,
2 2P x σ
μ − LSL
.
C pl =
3 2(1 − P x )σ

9.2.4 Clements’ method
4
Clements replaced 6σ in equation (9.1) by U p − L p :
USL − LSL
C p = ,
U p − L p
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where U p is the 99.865 percentile and L p is the 0.135 percentile. For C pk , the process
mean μ is estimated by the median M, and the two 3σs are estimated by U p − M and
M − L p respectively, giving

USL − M M − LSL

C pk = min , .
U p − M M − L p
Clements’ approach uses the classical estimators of skewness and kurtosis that are
based on third and fourth moments respectively, which may be somewhat unreliable
for very small sample sizes.