# the 3.4 number….

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- March 10, 2001 at 5:00 am #27113
I’m struggling to understand from where the 3.4 number is coming from! All the time I’ve been looking for an explanation of 6 sigma I’ve mostly found:

“The objective of Six Sigma Quality is to reduce process output variation so that ±six standard deviations lie between the process specification upper and lower limits. This will allow no more than 3.4 defect Parts Per Million (PPM) opportunities, also known as Defects Per Million Opportunities (DPMO), to be produced.”My question is, from where is the 3.4 coming from? where can I find a full explanation?

Thanks a lot for you help,

Dave0March 10, 2001 at 5:00 am #65945The term six sigma is a bit of a misnomer. A 1.5 sigma zone is constructed about the mean to allow for undetected process drift over time. The 1.5 sigma shift is highly controversional, but that is another subject. In any case, 3.4 defects per million opportunities is calculated from the tail area of a nornmal distribution beyond 4.5 sigma. The idea is to reduce sigma, assuming fixed specifications, such that the distance to the spec limit is at least 4.5 sigma — again assuming the 1.5 sigma shift. Six Sigma probably would sell very well if it were called Four and a Half Sigma. Hope this brief discussion is of value.

0March 10, 2001 at 5:00 am #65946

Neil PolhemusParticipant@Neil-Polhemus**Include @Neil-Polhemus in your post and this person will**

be notified via email.Fact: 3.4 out of a million is the probability that an observation randomly sampled from a normal distribution will be more than 4.5 standard deviations above the mean. Assumption: measurements taken from your process behave like random samples from a normal distribution. Assumption: the long-term mean of your process is at least six standard deviations away from any spec limit (hence “Six Sigma”). Assumption: the short-term mean of your process is never more than 1.5 standard deviations away from the long-term mean. Conclusion: no more than 3.4 out of a million measurements will be beyond the spec. In practice, validation of those assumptions is critical, including normality, amount of drift in the mean, randomness of deviations around that mean, and an accurate enough measurement process that meeting the “Six Sigma” criteria actually says something about the true state of your process.

0March 10, 2001 at 5:00 am #65947

Neil PolhemusParticipant@Neil-Polhemus**Include @Neil-Polhemus in your post and this person will**

be notified via email.“Four and One-Half Sigma” it is, although I might modify that slightly. Since the 3.4 out of a million is a one-tailed probability, a process with both an upper and a lower spec, each 4.5 standard deviations from the mean, would have a DPMO of 6.8, even if it never drifted away from the mean. “Four Point Six Five Sigma” would hold the two-tailed DPMO at 3.4. But then again, whose process is ever in a perfect state of statistical control?

0March 10, 2001 at 5:00 am #65948Neil makes a very good point. The 3.4 value does assume that the process may drift in one direction a maximum of 1.5 sigma. The one direction drift assumed in the Six Sigma calculation is another subtlety that is rarely explained or mentioned. As Neil writes there are a number of nontrivial assumptions that must be validated in order for the 3.4 number to be appropriate in the first place (see Neil’s very well written posting).

Personally, I am skeptical of the assumed 1.5 sigma shift and due to the considerable assumptions made in the 3.4 calculation, I do not emphasize 3.4 as a long term target for a quality improvement program. Of course this is controversial and there are valid opinions on both sides of the 1.5 sigma shift and 3.4 issue. BTW, there is no actual data or econometric model that determines 3.4 to constitute worldclass performance, it is expert opinion — I am not demeaning the proponents of 3.4 and their wisdom, I am simply stating that 3.4 is not based upon a quantitative assessment. Worldclass performance depends on too many factors to simply accept that an estimated 3.4 DPMO is the passport to the business equivalent of nirvana — it may be the passport to bankruptcy in some cases. Oh Well, I am getting a little too controversial here and really don’t wish to be!

I think Neil’s precise 4.65 sigma calculation (this is the technically correct sigma quality level one should shoot for in order to be a true 3.4 world class company) could lead to the creation of a 4.65 Sigma revolution in quality. These days one never knows!0March 11, 2001 at 5:00 am #65959Dave-

Joe’s numbers are correct. However, I have read rumoured stories that the 1.5 shift at Motorola came from the “ring” to the slogan “Six Sigma”. The story is that execs at Motorola new they needed to increase their quality levels to around a few defects per million to compete with the Japanese. 4.5 sigma didn’t have the slogan power that 6 did. They felt that they didn’t need the actual levels of 6 sigma quality however. I read the story in an ISSSP newsletter last fall. Check it out. Besides, how could you assume a 1.5 shift for all of your processes…does that make sense?0March 12, 2001 at 5:00 am #65975It is real simple. 3.4 is the number associated with one tail of the normal curve at 4.5 (6 sigma process shifted 1.5).

Is 1.5 appropriate. Simply put — yes, unless you have better data and if you do please share it.

0March 12, 2001 at 5:00 am #65981What makes you think that something in the ISSSP newletter is right? No one there was at Motorola.

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