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Finding a flat or not modeling distribution: how to manage it?

Every process is multi-modal, based on the Total Variance Equation. Tool wear - the predominate variation in precision machining - is a continuous uniform distribution, but most other errors, such as sampling error, gage error, etc. are normal or otherwise tailed. Add them all up and you get a predominately uniform distribution with small tails contributed by the other factors in the equation.
So this issue should be the reason to stay on 75% spec. values? Then the combination of distributions, getting uniform behavior with tails on the extreme sides?


Stop X-bar/R Madness!!
So this issue should be the reason to stay on 75% spec. values? Then the combination of distributions, getting uniform behavior with tails on the extreme sides?
Exactly. For you, as you know, those variations will be sampling, material thickness, physical properties and temperature, as examples. Most of those variations will be normal. Temperature may be a skewed distribution, if it reaches a steady state at some point in the run.
I don't know (I didn't found till today a clear case like that, about a continuous drift due to wearing) if it's a wearing situation too on stamping, but looking at a control chart, if set on 75% of spec. dimensions, if for a batch the mean shift on, e.g., 5% to upper limit control (that is set to 75% spec.), the chart tells that the process is going however ok (not considering the Western rules), but who tells me that the distribution has no tails going out of spec.? It's not a uniform distribution, you don't find it, it could be normal or not, largest extreme values, etc., the capability in that situation could easily be less than 1.33.
I didn't compared machining process to stamping process, you did it, describing stamping a long-wearing process like the machining one!
So, I could understand and theoretically agree with your assumption on precision machining (uniform distribution+tails of other error noises), but if you get a different distribution, the situation is not like that! So, thinking of it, the uniform distribution could be easier to manage than normal or worse non-normal, with tails to manage!

Bev D

Heretical Statistician
Staff member
Super Moderator
Tails and 1.33/75%.

All real processes have some amount of tails. NO process fits any theoretical distribution. The problem with the theoretical distribution is that we don't know how large the tails really are. That is why they are almost always 'larger' in the theoretical distribution than in the real process distribution.
The 'goal' of a 1.33 Cpk/Ppk was originally simple: to have reduced variation about the target. Somewhere along the line this got conflated with theoretical defect rates based on a theoretical distribution. Bob correctly recommends 75% of the tolerance so that the process has reduced variation and it allows for some of the tail of a real world '~ uniformly distributed' process.

Sullivan, L. P., “Reducing Variability: A New Approach to Quality”, Quality Progress, July 1984 and “Letters” Quality Progress, April, 1985

Leonard, James, “I Ain’t Gonna Teach It”, Process Improvement Blog, 2013 "I ain't gonna teach it!" - Jim Leonard - Process Improvement

Wheeler, Donald, “Probability Models do not Generate Your Data”, Quality Digest, March, 2009 Probability Models Don’t Generate Your Data | Quality Digest

Pyzdek, Thomas, “Why Normal Distributions Aren’t (All That Normal)”, Quality Engineering 1995, 7(4), pp. 769-777 Available for free as “Non-Normal Distributions in the Real World” at Non-Normal Distributions in the Real World | Quality Digest

Gunter, Berton H., “The Use and Abuse of Cpk”, Statistics Corner, Quality Progress, Part 1 January 1989, Part 2 March 1989, Part 3 May 1989, Part 4 July 1989
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