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August 31, 2006
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Illustration of the Two Methodologies Using a Case Study Data set 133
10 620 USL
10 865 Process Data
11 360 LSL
11 380 Target
11 445 USL 91.00000
11 540 Sample Mean 12.61467
11 560 Sample N 30
11 610 StDev (Within) 1.38157
StDev (Overall) 1.38157

13.980
14.380
14.410
15.225
15.335
16.300 11 22 33 44 55 66 77 88

Figure 10.1 Histogram of case study data.

specification limit is so far away from the mean and we do not expect USL to be
exceeded.
As the distribution is highly skewed, a normal approximation will not yield a good
estimate. Two methods were used to estimate the capability of this process; these are
considered in turn in Sections 10.2.1 and 10.2.2 and compared in Section 10.2.3.

10.2.1 Estimation of process capability using Box--Cox transformation
Traditionally, when the distribution of the data is not normal, the advice is to first
transform the data using the Box--Cox transformation. If the transformation is suc-
cessful, and the transformed data follows the normal distribution as proven by the
normality test, then we can proceed to do a PCA using the transformed data and the
transformed specification using formula (10.1).

10.2.1.1 Box--Cox transformation
The Box--Cox transformation is also known as the power transformation. It is done
by searching for a power value, λ, which minimizes the standard deviation of a stan-
λ

dardized transformed variable. The resulting transformation is Y = Y for λ = 0 and
Y = ln Y when λ = 0. Normally, after the optimum λ has been determined, it is then

rounded either up or down to a number that make some sense; common values are
1
1
−1, − / 2 ,0, / 2 , 2, and 3.
The transformation can be done by most statistical software. Figure 10.2 was ob-
tained from MINITAB 14. It shows a graph of the standard deviation over the different
values of λ from −5 to 5. The optimum λ (corresponding to lowest standard devia-
tion) suggested was −3.537 25. It is recommended to choose λ =−3; it is evident from
Figure 10.2 that this gives a standard deviation close to the minimum.