# Box-Cox transformation

Box-Cox Transformation
Performs a Box-Cox procedure for process data used in control charts. To use Box-Cox, the data must be positive.

 Box-Cox Transformation
When you ask Minitab to estimate lambda, you get graphical output. See Box-Cox Transformation − Graphical Output for an explanation of the output.

The Box-Cox transformation can be useful for correcting both nonnormality in process data and subgroup process variation that is related to the subgroup mean. Under most conditions, it is not necessary to correct for nonnormality unless the data are highly skewed. Wheeler [31] and Wheeler and Chambers [30] suggest that it is not necessary to
transform data that are used in control charts, because control charts work well in situations where data are not normally distributed.
They give an excellent demonstration of the performance of control charts when data are collected from a variety of nonsymmetric distributions.

Minitab provides two Box-Cox transformations: a standalone command, described in this section, and a transformation option provided with all control charts, except the Attributes charts. You can use these procedures in tandem. First, use the standalone command as an exploratory tool to help you determine the best lambda value for the transformation. Then,
when you enter the control chart command, use the transformation option to transform the data at the same time you draw the chart.

Dialog box items
All observations for a chart are in one column: Choose if data are in one or more columns, then enter the columns.
Subgroup sizes: Enter a number or a column of subscripts.
Observations for a subgroup are in one row of columns: Choose if subgroups are arranged in rows across several columns, then enter the columns. <Options>

Data − Box-Cox Transformation
Use this command with subgroup data or individual observations. Structure individual observations down a single column. Structure subgroup data in a single column or in rows across several columns − see Data for examples. When you include or exclude rows using control chart options > Estimate, Minitab only uses the non-omitted data to find lambda.

To do a Box-Cox transformation

Example of Box-Cox transformation
The data used in the example are highly right skewed, and consist of 50 subgroups each of size 5. If you like, you can look at the spread of the data both before and after the transformation using Graph > Histogram.

1 Open the worksheet BOXCOX.MTW.
 Box-Cox transformation-1
2 Choose Stat > Control Charts > Box-Cox Transformation.

 Box-Cox transformation
3 Choose All observations for a chart are in one column, then enter Skewed.

 Box-Cox transformation-3
4 In Subgroup sizes, enter 5.
 Box-Cox transformation-4
5 Click Options. Under Store transformed data in, enterC2. Click OK in each dialog box.

 Box-Cox transformation chart
Interpreting the results
The Lambda table contains an estimate of lambda (0.039501) and the best value
(0.000000), which is the value used in the transformation. The Lambda table also includes the upper CI (0.292850) and lower CI (−0.207308), which are marked on the graph by vertical lines.

Although the best estimate of lambda is a very small negative number, in any practical situation you want a lambda value that corresponds to an understandable transformation, such as the square root (a lambda of 0.5) or the natural log (a lambda of 0). In this example, 0 is a reasonable choice because it falls within the 95% confidence interval. Therefore, the natural log transformation may be preferred to the transformation defined by the best estimate of lambda.

The 95% confidence interval includes all lambda values which have a standard deviation less than or equal to the horizontal line. Therefore, any lambda value which has a standard deviation close to the dashed line is also a reasonable value to use for the transformation. In this example, this corresponds to an interval of − 0.207308 to 0.292850.

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