3. Production Process Characterization
3.4. Data Analysis for PPC

## Checking Assumptions

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 non-normal, such as:
• The data come from two or more different sources. This type of data will often have a multi-modal distribution. This can be solved by identifying the reason for the multiple sets of data and analyzing the data separately.
• The data come from an unstable process. This type of data is nearly impossible to analyze because the results of the analysis will have no credibility due to the changing nature of the process.
• The data were generated by a stable, yet fundamentally non-normal mechanism. For example, particle counts are non-normal by the very nature of the particle generation process. Data of this type can be handled using transformations.
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:

$$Y^{(\lambda)} = \left\{ \begin{array}{ll} y^{\lambda} & \mbox{if} \hspace{.2in} \lambda \ne 0 \\ \ln(y) & \mbox{if} \hspace{.2in} \lambda = 0 \end{array} \right.$$

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:

$$\ln(y), \hspace{.2in} \sqrt{y}, \hspace{.2in} \frac{1}{y}$$
General algorithm for trying to make non-normal data approximately normal The general algorithm for trying to make non-normal data appear to be approximately normal is to:
1. Determine if the data are non-normal. (Use normal probability plot and histogram).
2. Find a transformation that makes the data look approximately normal, if possible. Some data sets may include zeros (i.e., particle data). If the data set does include zeros, you must first add a constant value to the data and then transform the results.
Example: particle count data As an example, let's look at some particle count data from a semiconductor processing step. Count data are inherently non-normal. 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 log-transformed data and the assumptions will be approximately satisfied.
The original data is non-normal, the log transform looks fairly normal
Neither the square root nor the inverse transformation looks normal