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JWBK119-09
112 Computing Process Capability Indices for Nonnormal Data
This is the Johnson S L distribution that covers the lognormal family. The required
estimates for the parameters are
−1
x 0.95 − x 0.5
ˆ η = 1.645 log , (9.6)
x 0.5 − x 0.05
1 − exp (−1.645/ˆη)

*
ˆ γ = ˆη log , (9.7)
x 0.5 − x 0.05
*
ˆ ε = x 0.5 = exp − ˆγ /ˆη , (9.8)
where the 100αth data percentile is obtained as the α(n + 1)th-ranked value from n
observations. If necessary, linear interpolation between consecutive values may be
used to determine the required percentile.

9.2.6.2 The unbounded system (S U )
x − ε
τ 2 (x; ε, λ) = sinh −1 , −∞ < x < ∞. (9.9)
λ
Curves in the S U family are unbounded. This family covers the t and normal distribu-
17
tions, among others. For the fitting of this distribution, Hahn and Shapiro gave tables
for the determination of ˆ and ˆη based on given values of kurtosis and skewness.
γ
9.2.6.3 The bounded system (S B )

x − ε
τ 3 (x; ε, λ) = log , ε ≤ x ≤ ε + λ. (9.10)
λ + ε − x
The S B family covers bounded distributions, which include the gamma and beta dis-
tributions. Since the distribution can be bounded at either the lower end (ε), the upper
end (ε + λ), or both, this leads to the following situations.

Case I. Range of variation known. For the case where the values of both endpoints are
known, the parameters are obtained as

z 1−α − z α

ˆ η = (x 1−α −ε)(ε+λ−x α ) , (9.11)
log
(x α −ε)(ε+λ−x 1−α )

x 1−α − ε

ˆ γ = z 1−α − ˆη log . (9.12)

ε + λ − x 1−α
Case II. One endpoint known. In this case an additional equation obtained by matching
the median of the data is needed to supplement equations (9.11) and (9.12). This
equation is given by

ˆ λ = (x 0.5 − ε) (x 0.5 − ε)(x α −ε) + (x 0.5 −ε)(x 1−α −ε) −2 (x α − ε)(x 1−α − ε)
−1

2
× (x 0.5 − ε) − (x α − ε)(x 1−α − ε) .