The AD Normality P-Value Transformed
Data value of 0.404 confirms that the Box-Cox
transformation to normality was successful. The process capability
indices and expected performance can now be used to establish a
baseline performance. Note that there are no out-of-control signals
on the control charts, so the signals observed earlier when
normality was assumed were false alarms.
The Individuals – Original Data chart displays the untransformed data with control limits calculated as:
UCL = 99.865 percentile
CL = 50th percentile
LCL = 0.135 percentile
The benefit of displaying this chart is that one can observe the original untransformed data. Since the control limits are based on percentiles, this represents the overall, long term variation rather than the typical short term variation. The limits will likely be nonsymmetrical.
The Individuals/Moving Range – Normalized Data chart displays the transformed z-values with control limits calculated using the standard Shewhart formulas for Individuals and Moving Range charts. The benefit of using this chart is that tests for special causes can be applied and the control limits are based on short term variation. The disadvantage is that one is observing transformed data on the chart rather than the original data.
Automatic Best Fit
Now we will redo the capability analysis using the Automatic Best Fit option.
The 2 Parameter Loglogistic distribution was
selected as the best fit distribution. For details on how this
selection was made, see
Appendix: Statistical Details for
Nonnormal Distributions and Transformations.
The Anderson Darling statistic for the Loglogistic distribution is 0.245 which is less than the 0.37 value for the AD Normality test of the Box-Cox transformation indicating a better fit. (Note that published AD p-values for this distribution are limited to a maximum value of 0.25. The best fit selection uses a p-value estimate that is obtained by transforming the data to normality and then using a modified Anderson Darling Normality test on the transformed data).Another helpful tool to evaluate transformations and distributions is Distribution Fitting.