Page 156 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

August 31, 2006
2:57
Char Count= 0
JWBK119-10
A Further Case Study 141
10.2.3 Comparison of Results
The estimation of the process capability, C pk , is 1.19 using the Box--Cox transformation
and 5.94 using the best-fit distribution. The difference is 4.75, which is huge. Of course,
all estimates are wrong, but some are more wrong than others. What is needed is an
estimate that is good enough for decision-making. On which process needs a 100 %
screening and which one just needs sampling for Xbar-R chart.
Looking at the data and specification limit for the case study, there is no room for
doubt that this is a highly capable process, much more so than the common industrial
standard for the C pk (1.33). Whether the C pk is 3 or 5 does not really matter as it does
not affect the decision. But using the Box--Cox transformation method alone would
lead us to suspect that the process is not capable enough. Relying on this method alone
would here led us to commit resources that ultimately would have been wasted.
The reason for the gross inaccuracy when the λ is negative lies in the fact that the
minimum value for the transformed data is 0, violating the assumption that normal
data can take any value from −∞ to +∞. This violation is not a big problem if the
mean of the transformed distribution is far away from zero. But if the mean is close
to zero (which in most cases it will be when the distribution is transformed with
a negative λ, as any value greater than one will be between zero and one after the
transformation), the estimation will think that the distribution has a ‘tail’ all the way
to −∞, and this will bring down the C pk estimate.

10.3 A FURTHER CASE STUDY

Another data set was taken from another process. The summary statistics and the
histogram are shown in Figure 10.8. The upper specification limit for this process is
75, and there is no lower specification limit.

Summary for Case2
Anderson-Darling Normality Test
A-Squared 1.33
P-Value < 0.005
Mean 19.403
StDev 2.724
Variance 7.421
Skewness 1.13045
Kurtosis 1.94975
N 100
Minimum 14.820
1st Quartile 17.481
Median 18.855
15 18 21 24 27 30 3rd Quartile 20.891
Maximum 29.811
95% Confidence Interval for Mean
18.863 19.944
95% Confidence Interval for Median
18.284 19.835
95% Confidence Intervals
95% Confidence Interval for StDev
Mean 2.392 3.164
Median
18.5 19.0 19.5 20.0
Figure 10.8 Summary statistics for Case Study 2.