Gaging Improves Automatic Grinder Control
I would like to comment on a statement in the article "Gaging Improves Automatic Grinder Control" in April 2002 Quality. The statement deals with the Cpk and is found in the first paragraph of page 34. The statement is:
"A Cpk of 2.0 means that 68% of the parts will be +/- 33 millionths of an inch of nominal."
1. I assume it meant to say " -- will be WITHIN +/- 33 millionths of an inch of Nominal."
2. The Cpk has little, if anything at all, to do with the Nominal value of a process. It is the relative distance the closest spec limit is to the average value. If we assume, for example, that the Upper Spec Limit is the one which is closest to the average value, then the Cpk would be calculated as: Cpk = (USL - Average) / (3*Std Deviation). I believe it is clear that the Nominal value is nowhere in the definition.
3. However, trying to rationalize "68% are within 33 millionths", let's assume:
A1. the average is actually equal to the Nominal and
A2. the Nominal is half way between USL and LSL. If so, then the cpk becomes: (USL -(USL+LSL)/2) / (3*Std Deviation) = (USL - LSL) / (6-Std Deviation) which is also, of course, the Cp.
4. Next, the 68% seems to have a ring of 1 standard deviation about it. So lets see where that leads us. The Cpk is given as 2, thus 2 = (USL - LSL) / (6*Std Deviation), and USL - LSL is given as 0.0002. Thus the standard deviation must have been 0.0002/6 = 0.000330. Eureka! 68% should be within 0.000033 inches of Nominal.
5. This result is based on A1 and A2. Of those two assumtions, A2 is quite common but A1 is process dependent and really only rarely true.
6. If the average value is not the Nominal but, say for example, Nominal plus one standard deviation, one could solve for a value of the standard deviation which would still yield a Cpk of 2 - but "within +/- of" would need modification.
7. To summarize: Statements which include both "Nominal" and "Cpk" should be avoided. The Cpk is a confounded quality measure - one which confounds process center and process width.
Mr. Holmes is correct. The sentence should read "A Cpk of 2.0 means that 68% of the parts will be WITHIN +/- 33 millionths of an inch of nominal."