Page 164 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

Char Count= 0
August 31, 2006
2:57
JWBK119-10
References 149
Cpk estimation
6 BoxCox
Best-Fit
5
Delta
Poly. (Delta)
4
Power (BoxCox)
Power (Best-Fit)
3
Cpk
2
1
0
0 5 10 15 20
−1
USL (# of Std_Dev away from mean)

Figure 10.18 Comparison of process capability between Box-Cox with power of −1 vs. Log-
normal fit.

10.4.2 Comparison of results

It can be seen from Figure 10.18 that the difference in the estimation of C pk widens
as the process grows more capable. When C pk < 1, the difference is negligible. When
the process capability (as estimated by the Box--Cox method) is more than 2, although
the difference is very big, this does not really matter in real life as it does not affect
the decision, that is the process is very capable and therefore no effort is required to
improve it. The critical region is between them where the difference can change the
decision from ‘leave it alone’ to ‘need to improve the process’.

10.5 SUMMARY

When Box--Cox transformation is used in estimation of C pk for a distribution that is
skewed to the right, as compared to a reasonable fit using the best-fit distribution,
it is likely to be characterized by underestimation. The higher the process capability,
the worse the underestimation. When the C pk is low (below 1), estimation using both
methods is very close. Therefore, it is advisable not to use Box--Cox transformation
for the purpose of process capability analysis when λ is negative.
Today, most statistical software (e.g. MINITAB 14) allows user to select the best-fit
distribution for process capability analysis. This makes PCA for non-normal data an
easy task without using the Box--Cox transformation.

REFERENCES

1. D’Agostino, R.B. and Stephens, M.A. (1986) Goodness-of-Fit Techniques. New York: Marcel
Dekker.
2. Chambers, J., Cleveland, W., Kleiner, B. and Tukey, P. (1983) Graphical Methods for Data Anal-
ysis. Belmont, CA: Wadsworth.