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JWBK119-10
134 Process Capability Analysis for Non-Normal Data with MINITAB
Box-Cox Plot of Data
Upper CL
Lambda
1.9
(using 95.0% confidence)
Estimate −3.53725
1.8 Lower CL
Upper CL −0.50142
1.7
Best Value −3.53725
StDev 1.6
1.5

1.4
1.3 Limit

1.2
−5.0 −2.5 0.0 2.5 5.0
Lambda
Figure 10.2 Box--Cox plot for case study data.

10.2.1.2 Verifying the transformed data
Having transformed the data, we need to check that the transformation is a good one
using both a normality test and visual assessment (fitting a normal curve over the
histogram).
There are many normality tests available. The Anderson--Darling test is the test
1
recommended by Stephens. It tests the null hypothesis that the data come from a
specified normal distribution by measuring the area between the fitted line based
on the distribution and the nonparametric step function based on the plotted points.
The statistic is a squared distance and is weighted more heavily in the tails of the
distribution. It is given by
1
2
A =−N − (2i − 1)(ln F(Y i ) + ln(1 − F(Y N+1−i )),
N
where the Y i are the ordered data and

Y i − ¯x
F(Y i ) =
s
is the cumulative distribution function of the standard normal distribution. Smaller
Anderson--Darling values indicate that there is are fewer differences between the data
and the normal distribution, and hence that the data fits the specific distribution better.
Another quantitative measure for reporting the result of the Anderson--Darling
normality test is the p-value, representing the probability of concluding that the null
2
hypothesis is false when it is true. If you know A you can calculate the p-value. Let

0.75 2.25
2 2
A = A * 1 + + 2 ,
N N