Friday, March 13, 2009

More Prevaricating about Sigma

The one anonymous comment I received the other day after my initial posting seemed to imply that I am somehow against Six Sigma. Nothing could be further from the truth. I am currently a Quality consultant, and a large part of my business has been (and continues to be) Six Sigma. My background goes beyond Six Sigma, though, and I do have some reservations about some of the common practices within Six Sigma.
Before I ever heard of Six Sigma, I was heavily involved in the Navy’s Total Quality Leadership initiative. I had studied all the quality gurus, consulted and written courses in Statistical Process Control and systems approaches to process improvement, taken a masters degree in Quality and applied statistics. I had been director of quality for a large overseas base and an internal consultant to the entire Department of the Navy.
From the viewpoint of statistical methods, I initially viewed Six Sigma with some suspicion. There was the matter of the “1.5 Sigma Shift” applied blindly to all the calculations. This is just For one thing, one of the first slides in the deck, in the first presentation I saw, was a direct copy of one I had seen at a Crosby presentation. It talked about what you might get in “a three-sigma world,” and listed the usual “Babies dropped on heads” and “Airplane crashes,” etc. This is a great marketing slide, especially when coupled with the one that inevitably follows it, showing the several-order-of-magnitude improvement in “a six-sigma world.” The Crosby presentation, of course, followed the initial slide with a slide showing the improvement in a “zero-defects world.”
This slide is a pretty persuasive slide. It deals with errors that we would, of course, want to get as close to zero as possible. We can’t tolerate any aircraft falling from the sky, and can’t tolerate any babies being dropped on their heads, so of course three sigma (Cpk = 1) is never going to be as good as six sigma (Cpk = 2), but it’s somewhat disingenuous, for a number of reasons:
The calculations are correct, unless you are using the common “1.5-Sigma Shift,” in which case the six-sigma world calculations are too pessimistic (more on the shift later).
The idea of three sigma and six sigma come from the world of Statistical Process Control (SPC) and Capability Studies for continuous data, where you have tolerance limits (usually two-sided), around a process average. The data in the slide are for errors, countable things, discrete data, and all the examples are about the types of errors where the only acceptable tolerance limit is the lower bound of zero. Normal distribution theory doesn’t usually apply in this situation.
I’ve never understood why three sigma was the starting point for these slides, but I’ve always suspected that it was to promote the misconception that SPC somehow “stops” at three sigma (because of the three-sigma control limits used in SPC), and so the other approaches touted in the second slide are superior. Nothing could be further from the truth; if you watch or read “The Japanese Control Chart” by Don Wheeler, you’ll see a Japanese company that gets to TWELVE sigma, just using a paper control chart.
So I don’t really like that slide, but it worked OK as a marketing tool. From the viewpoint of statistics, I worried mostly about the mixing of the methods used to assess DPMO (Defects per Million Opportunities) and the 1.5-sigma shift. An excellent paper by Roger Hoerl got me over that. He pointed out that traditional views of capability left out the idea of mistakes or defects, and made a strong case for a metric like DPMO that can translate the idea of capability across distributions.
The “1.5 Sigma Shift” is another can of worms, but not for statisticians. Put simply, statisticians pay very little attention to it. It’s just a transformation that we will probably all be stuck with until someone with enough credibility to stop a flawed practice yells “STOP!” It’s relatively harmless, anyway, and it works as a “fudge factor” or a safety factor so you generally end up beating expectations. While it is true that undetected shifts in even a well-controlled process might allow for a lot of closer-to-spec product to be produced over many days, that does not justify an arbitrary value of 1.5 sigma to be universally applied. There are a lot of reasons for this…we’ll get into it another time.

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