## Parts Per Million

In the Motorola process, as practiced, the process mean is allowed to drift up to 1.5s in either direction, and the process random effects should stay within an additional ±3s from the mean at all times. Thus, in the limit, the engineering tolerance must allow for a total of ±4.5s from the mean. At ±4.5s , the confidence is so high that speaking in percentages, as we have done to this point, is very awkward. Therefore, one interesting contribution made by the promoters of Six Sigma was to move the conversation of confidence away from the idea of percentages and toward the idea of "errors per million opportunities for error." At the limit of ±4.5s , the confidence level in traditional form is 99.9993198% or, as often said in engineering shorthand, "five nines."

However, in the Six Sigma parlance, the process engineers recognize that the tails of the Normal distribution, beyond 4.5s in both directions, hold together only 6.8 * 10-6 of the area under the Normal curve:

Total area under the Normal curve = 1

Each tail, being symmetrical on each side, holds only 3.4 * 10-6 of the area as shown in Figure 8-9. t8] Thus, the mantra of the Six Sigma program was set at having the engineering tolerance encompass all outcomes except those beyond 4.5s of the mean. In effect, the confidence that no outcome will occur in the forbidden bands is such that only 3.4 out-of-tolerance outcomes (errors) will occur in either direction for every 1 million opportunities. The statement is usually shortened to "plus or minus 3.4 parts per million," with the word "part" referring to something countable, including an opportunity, and the dimension that goes with the word "million" is silently implied but of course the dimension is "parts."

Standard Normal Distribution

Standard Normal Distribution