3.
Production
Process Characterization
3.4. Data Analysis for PPC


Check the normality of the data  Many of the techniques discussed in this chapter, such as hypothesis tests, control charts and capability indices, assume that the underlying structure of the data can be adequately modeled by a normal distribution. Many times we encounter data where this is not the case.  
Some causes of non normality 
There are several things that could cause the data
to appear nonnormal, such as:


We can sometimes transform the data to make it look normal 
For the last case, we could try transforming the data using
what is known as a power transformation. The power
transformation is given by the equation:
where Y represents the data and lambda is the transformation value. Lambda is typically any value between 2 and 2. Some of the more common values for lambda are 0, 1/2, and 1, which give the following transformations:


General algorithm for trying to make nonnormal data approximately normal 
The general algorithm for trying to make nonnormal data
appear to be approximately normal is to:


Example: particle count data  As an example, let's look at some particle count data from a semiconductor processing step. Count data are inherently nonnormal. Below are histograms and normal probability plots for the original data and the ln, sqrt and inverse of the data. You can see that the log transform does the best job of making the data appear as if it is normal. All analyses can be performed on the logtransformed data and the assumptions will be approximately satisfied.  
The original data is nonnormal, the log transform looks fairly normal  
Neither the square root nor the inverse transformation looks normal 