Why Six Sigma?

Anyone looking at a table of probabilities for the normal (Gaussian) distribution will wonder what six-sigma has to do with 3.4 defects per million things. Only one billionth of the normal curve lies beyond six standard deviations, or two billionths if you count both too-high and too-low values. Conversely, a mere three sigma corresponds to just 2.6 problems in a thousand, which would seem a good result in many businesses.

The answer has to do with practical considerations for manufacturing processes. (The following discussion is based loosely on the treatment by Robert V. Binder in a discussion of whether six-sigma practices can apply to software [1] (http://www.rbsc.com/pages/sixsig.html).) Suppose that the tolerance for some manufacturing step (perhaps the placement of a hole into which a pin must fit) is 300 micrometres, and the standard deviation for the process of drilling the hole is 100 micrometres. Then only about 1 part in 400 will be out of spec. But in a manufacturing process, the average value of a measurement is likely to drift over time, and the drift can be 1.5 standard deviations in either direction. At any time, 6.6% of the output will be off by 1.5 sigma in each direction. Thus, when the process has drifted by 150 micrometres, 6.6% of the product will be off by 150 + 150 or 300 micrometres, and therefore out of spec. This is a high defect rate.

If you set the tolerance to six sigma, then a drift of 1.5 sigma in the manufacturing process will still produce a defect only for parts that are more than 4.5 sigma away from the average in the same direction. By the mathematics of the normal curve, this is 3.4 defects per million.

The 1.5 sigma shift is very problematic, to say the least.

Common practice is to represent a truly 4.5 sigma process as a 6 sigma process. This is the reverse of what you would expect, if you were "derating" your sigma number, to account for unobserved, but expected variation. If you were doing that, you would represent a 4.5 sigma process as a 3 sigma process, not a 6 sigma process.

It has been suggested that one of the early practitioners of six sigma invented or adopted the 1.5 sigma shift purely for marketing reasons. It was unrealistic to expect to reduce defect to the few parts per billion level, and he didn't want to sell a program named "4.5 Sigma", so a 1.5 sigma shift was necessary, to get an attractive name.

However, according to original training material and a handout dated 1985 from Motorola, Six Sigma is actually a Cpk of 1.5 and a Cp of 2.0. Based on a 1200 parts/step process, and using a 3 sigma design margin, ‘fewer than 4 units out of every 100 would go through the entire manufacturing process without a defect’ and thus, we can see that for a product to be built virtually defect-free, it must be designed to accept characteristics which are significantly more than +/- 3 sigma away from the mean.

'A design specification width of +/- 6 Sigma and a process width of +/- 3 Sigma yields a Cp of 12/6 = 2. However, the process mean can shift. When the process mean is shifted with respect to design mean, the Capability Index, (Cp), is adjusted with a factor k, and becomes Cpk.' The important difference here is Design vs. Process.

Nonetheless it is the case that processes drift over time due to noise factors, and a shift of +/-1.5 standard deviations is the limit at which the shift becomes detectable with a sample size of 4, prompting investigation of an "out of control" process.

When many parts have to fit together, tolerances actually work in the favor of the manufacturer. It is quite possible to make six sigma assemblies out of three sigma parts, since it is highly unlikely that all parts will simultaneously be at one extreme of the tolerance range. Intelligently allocating variation is called "Statistical Tolerancing", and is a useful part of Design for Six Sigma.

Clearly, many things on which people rely (services, software products, etc.) are not manufactured by machine tools to particular measurements. In these cases, "six sigma" has nothing to do with statistical distributions, but refers to a goal of very few defects per million, by analogy to a manufacturing process. The usefulness of the analogy is controversial among those concerned with quality in non-manufacturing processes.