6.5.2. What do we do when data are "Non-normal"?

6. Process or Product Monitoring and Control
6.5. Tutorials

6.5.2. What do we do when data are "Non-normal"?

Often it is possible to transform non-normal data into approximately normal data

Non-normality is a way of life. One strategy to make non-normal data resemble normal data is using a transformation. There are no dearth of transformations in statistics, the issue is which one to select for the situation at hand. Unfortunately the choice of the "best" transformation is not obvious.

This was recognized in 1964 by G.E.P. Box and D.R. Cox. They wrote a paper in which a useful family of power transformations was suggested. These transformations are only defined for positive data values. This should not pose any problem, because a constant can always be added if the set of observations contains one or more negative values.

The Box-Cox power transformations are given by

The Box-Cox Transformation

Given data observations x₁, x₂, ...x_n, the way to select the power l is to use the l that maximizes the logarithm of the likelihood function

The logarithm of the likelihood function

In addition a confidence interval can be constructed for l as follows:

A set of l values that represent an approximate 100(1-a) % confidence interval for l is formed from those l that satisfy

Example of the Box-Cox scheme

To illustrate the procedure we used the data from Johnson and Wichern's textbook (Prentice Hall 1988), Example 4.14. The observations are microwave radiation measurements
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The resulting values of the log likelihood function by varying l from -2.0 to 2.0 are given below

This table shows that l = .3 maximizes the log likelihood function. This becomes .28 if a second digit of accuracy is calculated.