One of the most common questions I get from students in my Process Capability Class is, how can I use the capability index from my process to approximate a defect level for my process?
For most, the answer comes from using a table like this one:
Using this table, we can see that a Cpk (or Ppk) of 1.33 equates to a +/-4 sigma process which yield approximately 63 defects per million parts produced.
To approximate the PPM for your specific Cpk level, you can shift over to Microsoft Excel. It’s easy to see from the table that the “sigma level” is 3 times the capability index (e.g. a Cpk of 1.67 equates to a 5 sigma process.)
To get from the sigma level to an approximate PPM in Excel, you use this formula:
where Z = the sigma level
Now the astute observer may already see a problem with this approach:
1-NORM.S.DIST(Z,TRUE) gets you to the cumulative portion of defective parts on one side of the curve, namely the side where the specification limit is closest to the arithmetic average of the process. Afterall, the Cpk and Ppk indexes only consider the specification limit closest to the process mean; the opposite side is discarded in the equation.
Multiplying this output by 2 assumes that both sides of the curve yield an equal portion of parts out of specification, resulting in an approximation of the portion defective. This however is a faulty assumption especially for processes that are dramatically shifted to one side or other of the specification range.
A more precise estimation of PPM would require your breaking up Cpk (or Ppk) into its constituents, Cpl and Cpu (Ppl and Ppu). Again, multiplying these indexes by 3 yields their individual sigma levels. Then summing the results of the NORM.S.DIST equations would yield that more precise PPM result:
PPM = [(1-NORM.S.DIST(Z,upper,TRUE)) + (1-NORM.S.DIST(Z,lower,TRUE))] * 1,000,000
For most applications, capability indexes like Cpk and Ppk are rounded to two decimal places. The error resulting from this rounding operation is often greater than the improvement in accuracy by using this two-sided equation. Nonetheless, it’s better to fully understand the assumptions baked into your equations, than to unknowingly allow errors into your work.
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