Process Capability

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    Mike Archer

    Hello.  I have been reading to find out why cp, cpk, pp, ppk all use 3 sigma to define the size of a process.  What is the significance in 3 sigma.  The general cutomer requirement is 4 sigma, so that translates to a cp of 1.33.  Wouldn’t it have been more straight forward for cp, cpk, etc to use 4 sigma and then an acceptable cp would be an even 1.0?  I will be teaching process capability next week and I really want to be able to explain what is so important about 3.  I have read several articles, and have not found an explanation.


    Erik L

    Capability is an opportunity to explore two key relationships, namely the Voice of the Customer (VOC) and the Voice of the Process (VOP).  There are multiple ways to explore this relationship, several of which you sighted.  VOC is typically expressed through the USL and LSL, or collapsed into the Tolerance for simplification, and the VOP is expressed through sample estimates of a processes average and standard deviation.  If we assume that the normal curve is an appropriate approximation of the data, we know that the majority of the data will fall +/- 3 standard deviations from the average (or the middle 99.73 percent of the processes output) and this is typically the limit that we stop in characterizing the spread in our data.  This is defined as having achieved minimum capability (no more than .135% beyond the specification limits).
    Achieving this basic level of quality was typically viewed as insufficient therefore traditional goals for capability analysis were to achieve a spread (6 standard deviations) equal to, or less than, 75% of the tolerance (expressed through Cr).  This would be the equivalent of 8 standard deviations equaling 100% of the tolerance.  Taking the inverse of the Cr goal gives you the standard goal for the Cp measure, and if we go with the same 75% goal, we get to our 1.333 capability goal for Cp.
    Cp, in effect, asks how many of these distributions (6 standard deviations across its base) could we potentially fit, side by side, within the associated tolerance.  A Cp of one would fit one distribution into the tolerance and a Cp of two would fit two etc. etc.  A Cp of 1.33 would give us 8 standard deviations spread within the limits of the tolerance. 
    So why do we divide by three in a Cpk calculation?  This was an attempt to simplify interpretation of the calculation when one achieved minimum capability.  If one would use Zmin as a calculation, just meeting the VOC (and obtaining no more than .135 % nonconforming beyond either specification limit) would equate to a score of three and obtainment of minimum capability requirement.  Cpk simplified the interpretation by adding the three in the denominator to make the obtainment of minimum capability equal to 1.  
    Well, hope that helped to address some of the questions that you had.


    The New MB

    Just  because   3 Sigma is  equal  to Cp= 1,below  that  the  process  is  incapable

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