How does normality of the data impact the process performance?

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First, there is a great deal of literature on this subject – find it and read it.  However, there is no substitute for direct experience.  Of course, this is directly related to the old saying “we learn by doing.”  Along these lines, I would strongly recommend that you further your Six Sigma training and education by way of computer simulation.  It is quite easy to set up a reoccurring Monte Carlo simulation in Excel.  Doing so will provide many answers (and insights) into your question – as well as those questions you have not yet asked.

For example, you can generate a random normal number with a mean of zero and a standard deviation of one with the simple algorithm: =NORMSINV(RAND()).  This algorithm transforms a random uniform number into a random normal number.  Of course, the algorithm can be scaled by: =M+(S*NORMSINV(RAND())), where M is the required mean and S is the desired standard deviation.  With this accomplished, you are free to create N number of such random normal numbers by way of the “copy down” feature in Excel.  Simply depress the”F9” key and the computer will generate a new set of N.  At this point, you are free to experiment and learn.  For example, let us suppose that you generate N random normal numbers and then take the square of each. 

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Of course, we would naturally seek to plot the two distributions in the form of histograms and then visually compare them.  From this exercise, it would be apparent that the transformed distribution can be “reset” to a normal distribution by merely taking the square root of each transformed value.  From simple mathematics we know this.  In this manner, we learn about transformations by reversing the process – we move from a normal distribution to a transformed distribution back to a normal rather than moving from a non-normal to a normal. 

With a little practice, you will be reshaping all kinds of distributions to meet your analytical needs.  The process of discovery will be staggering as your knowledge really begins to skyrocket.  Don’t forget to always correlate the original data to the transformed data to ensure that a high level of association still exists (following the transformation).  It is possible to transform your way to low correlation.  This point has huge implications when using transformed data to compute standard indices of capability, such as Cp, Cpk, Pp, and Ppk.

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