- Box-Cox Transformation (includes an automatic threshold option so that data with negative values can be transformed)
- Johnson Transformation
- Distributions supported:
- Half-Normal
- Lognormal (2 & 3 parameter)
- Exponential (1 & 2 parameter)
- Weibull (2 & 3 parameter)
- Beta (2 & 4 parameter)
- Gamma (2 & 3 parameter)
- Logistic
- Loglogistic (2 & 3 parameter)
- Largest Extreme Value
- Smallest Extreme Value
- Automatic Best Fit based on AD p-value

For technical details, see

Note that these transformations and distributions are particularly effective for inherently skewed data but should not be used with bimodal data or where the nonnormality is due to outliers (typically identified with a Normal Probability Plot). In these cases, you should identify the reason for the bimodality or outliers and take corrective action. Another common reason for nonnormal data is poor measurement discrimination leading to “chunky” data. In this case, attempts should be made to improve the measurement system.

- Open the file
**Nonnormal Cycle Time2.xlsx.**This contains continuous data of process cycle times. The Critical Customer Requirement is: USL = 1000 minutes. - Let’s begin with a view of the data using Histograms and Descriptive Statistics. Click
**SigmaXL > Graphical Tools > Histograms & Descriptive Statistics.** - Ensure that entire data table is selected. If not, check
**Use Entire Data Table**. Click**Next**. - Select
*Cycle Time (Minutes)*, click**Numeric Data Variable (Y)**>>. Click**OK**.

- We will confirm the nonnormality by using a Normal Probability Plot. Click
**Sheet 1**Tab (or**F4**). Click**SigmaXL > Graphical Tools > Normal Probability Plots**. - Ensure that the entire data table is selected. If not, check
**Use Entire Data Table**. Click**Next**. - Select
*Cycle Time (Minutes)*, click**Numeric Data Variable (Y)**>>. Click**OK**. A Normal Probability Plot of Cycle Time data is produced: - The curvature in this normal probability plot confirms that this data is not normal.
- For now, let us ignore the nonnormal issue and perform a
Process Capability study assuming a normal distribution. Click
**Sheet 1**Tab. Click**SigmaXL > Process Capability > Capability Combination Report (Individuals)**. - Select Cycle Time (Minutes), click
**Numeric Data Variable (Y)**>>. Enter**USL**= 1000; delete previous**Target**and**LSL**settings. -
Click
**OK**. The resulting Process Capability Report is shown below:

Notice the discrepancy between the Expected Overall (Theoretical) Performance and Actual (Empirical) Performance. This is largely due to the nonnormality in the data, since the expected performance assumes that the data is normal. So why not just use the actual performance and disregard the expected? This would not be reliable because the sample size, n = 30, is too small to estimate performance using pass/fail (discrete) criteria.

Also note that the process appears to be out-of-control on both the individuals and moving range charts.

- We will now perform a process capability analysis using the Capability Combination Report for Nonnormal Individuals. Click
**Sheet 1**Tab (or**F4**). Click**SigmaXL > Process Capability > Nonnormal > Capability Combination Report (Individuals Nonnormal)**. Ensure that the entire data table is selected. If not, check**Use Entire Data Table**. Click Next. - Select
*Cycle Time (Minutes)*, click**Numeric Data Variable (Y)**>>. Enter**USL**= 1000. We will use the default selection for**Transformation/Distribution Options: Box-Cox Transformation**with**Rounded Lambda**. Check**Tests for Special Causes**as shown: - Click
**OK**. The resulting Process Capability Combination report is shown below:

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**.

- Click
**Recall SigmaXL Dialog**menu or press**F3**to recall last dialog. Select**Automatic Best Fit**as shown: -
Click
**OK**. The resulting Process Capability Combination report is shown below. Please note that due to the extensive computations required, this could take up to 1 minute (or longer for large datasets):

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).

- Click
**Sheet 1**Tab (or**F4**). Click**SigmaXL > Process Capability > Nonnormal > Distribution Fitting**. Ensure that the entire data table is selected. If not, check**Use Entire Data Table**. Click**Next**. - Select
*Cycle Time (Minutes)*, click**Numeric Data Variable (Y)**>>. We will use the default selection for**Transformation/Distribution Options: All Transformations & Distributions**as shown: -
Click
**OK**. The resulting Distribution Fitting report is shown below. Please note that due to the extensive computations required, this could take up to 1 minute (or longer for large datasets):

The distributions and transformations are sorted in descending order using the AD Normality p-value on the transformed z-score values. Note that the first distribution shown may not be the selected “best fit”, because the best fit procedure also looks for models that are close but with fewer parameters.

The reported AD p-values are those derived from the particular distribution. The AD p-value is not available for distributions with a threshold (except Weibull), so the AD Normality p-value on the transformed z-score values is used (labeled as Z-Score Est.).

Since the sort order is based on the AD p-values from Z-Score estimates, it is possible that the reported distribution based AD p-values may not be in perfect descending order. However any discrepancies based on sort order will likely not be statistically or practically significant.

Some data will have distributions and transformations where the parameters cannot be solved (e.g., 2-parameter Weibull with negative values). These are excluded from the Distribution Fitting report.

The parameter estimates and percentile report includes a confidence interval as specified in the

The control limits for the percentile based Individuals chart will be the 0.135% (lower control limit), 50% (center line, median) and 99.865% (upper control limit). Additional percentiles may be entered in the

After reviewing this report, if you wish to perform a process capability analysis with a particular transformation or distribution, simply select

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