# Methods and formulas for Automated Capability Analysis

To select a reasonable method, Minitab Statistical Software evaluates multiple methods to find a reasonable fit for the data. From the order of evaluation, the analysis provides capability results for the first method that provides a reasonable fit to the data. To determine whether a method is a reasonable fit, the analysis uses an Anderson-Darling test with a significance level of 0.05.

The selection of the first method that provides a reasonable fit has advantages over a procedure that selects a method based on the best fit to the sample data. For example, the gamma distribution converges to a normal distribution as the shape parameter increases. Departures from normality will make the gamma distribution fit better than the normal distribution for some samples from a process that produces normally-distributed data. The normal distribution is a better method for a process that produces normally-distributed data even if another method fits better for a particular sample.

The analysis considers distributions first. The order of the distributions considers usefulness and practicality. For example, the normal distribution is important to evaluate because the central limit theorem shows why means of subgroups often follow a normal distribution. The following list gives the order of evaluation for the distributions:
1. Normal
2. Weibull
3. Lognormal
4. Smallest Extreme Value
5. Largest Extreme Value
6. Gamma
7. Logistic
8. Loglogistic
9. Exponential
10. 3-Parameter Weibull
11. 3-Parameter Lognormal
12. 3-Parameter Gamma
13. 3-Parameter Loglogistic
14. 2-Parameter Exponential

This order and the use of the Anderson-Darling test mean that some distributions are never automatic selections. The exponential distribution is a special case of the Weibull distribution, so the Weibull distribution will fit any data that fit an exponential distribution. If process knowledge is compatible with the characteristics of an exponential distribution, you can produce those capability results. Minitab Statistical Software does not produce a p-value for the Anderson-Darling test for distributions with a threshold parameter, except for the 3-parameter Weibull distribution and the 2-parameter exponential distribution. If process knowledge is compatible with a distribution like the 3-parameter lognormal distribution, you can produce those capability results.

Next, the analysis evaluates whether a transformation makes the data follow a normal distribution. Because transformations change the units of the data, the analysis automatically selects a transformation only if no distributions fit the data. If a transformation makes the data follow a normal distribution and you prefer the transformation results that include the within-process capability statistics, you can produce those capability results. The analysis tries the simpler Box-Cox transformation before the more complex Johnson transformation.

If no distribution fits the data and no transformation makes the data follow a normal distribution, then the analysis produces results for a nonparametric capability analysis.

If you add rows to the original columns for the analysis, you have the option to update the results or create new results in the output pane. When you select either of these options, the resulting capability results keep the same method, but the estimates of the distribution parameters update. With this approach, the addition of data provides capability results that are reasonable to compare to the previous results. The use of the same method provides clear information about how the new data changes the capability of the process. If the method does not provide a reasonable fit with the new data, consider whether the process is stable. Capability results for a process that changes characteristics do not sufficiently describe the latest state of the process.

For details on the methods and formulas for the capability statistics for a normal distribution or a transformation, go to Normal Capability Analysis.

For details on the methods and formulas for the capability statistics for a nonnormal distribution, go to Nonnormal Capability Analysis.

For the methods and formulas for a the capability statistics for a nonparametric analysis, go to Nonparametric Capability Analysis.

For details on the estimation of distribution parameters and the Anderson-Darling test, go to Individual Distribution Identification.