Six Sigma vs. Control Charts based on 3 Sigma Limits

10/27/2007:

The regular quality control chart usually uses 3 sigma for upper and lower limits.
The chances that the defect falls outside the limits are 0.26%, i.e. 99.74% of the output falls inside the limits. But if we look at the theory of six sigma, the defect rate at 3 sigma is 6.68% assuming normal distribution with upper and lower limits. Why such a big discrepancy?

Ed C.

We include the details of the Sigma Level calculations in our Black Belt course, and as you go through the Green Belt training course you will see some further explanation of how this differs from the control limits on a control chart.
A standard control chart with plus and minus three sigma limits is designed (assuming normality) to provide for a false alarm rate of 0.27%. That is, over the long run, we would expect 99.73% of the plotted statistic (i.e. the observations on an Individual-X chart, or the subgroup averages on an X-Bar chart) to lie within the control limits if the process is stable and the Normal distribution provides a good fit to the plotted statistic. Conversely, approximately 0.27% of the time (over the long term), subgroups will fall outside these limits when the process is stable (in-control), providing our estimate of the false alarm rate of the properly-designed control chart.
However, this false alarm rate cannot be compared to a Sigma Level for a number of reasons. First off, out of control is not synonymous with defective. A defect occurs when an observation exceeds a specification limit, usually defined by the customer. A subgroup out of control implies either a false alarm or a shift in the process. The control limits on the control chart are calculated based on the statistics of the plotted subgroups, and are not related to the specification limits. When we plot observations on the Individual X chart, it is only when the specifications coincide exactly with the calculated control limits that an out of control observation equates to a defective. On an X-Bar chart, we plot subgroup averages, so there is even less relationship. When an X-Bar chart is used, all of the observations in the subgroup may exceed the specification limits, yet the subgroup falls within the control limits. On an X-Bar chart, the control limits must be much narrower than the specification limits to achieve process capability. You will discover the reasoning for this in the X-Bar Charts topic of our training.
In the simple case of an Individual X chart where specification limits happen to exactly coincide with the calculated three sigma control limits, the estimated process defect rate for Six Sigma purposes is not 0.27 % (or 2700 DPMO), since over the longer term we accept that the process may shift by as much as 1.5 Sigma. This 1.5 Sigma shift is an "industry-standard" estimate of process Sigma Levels, as initially developed by Motorola and subsequently adopted throughout general industry applications. Using Appendix Table 8 in the Six Sigma Demystified text (or this DPMO to Sigma Level conversion), we see that the estimated process defect rate for the three sigma process over the longer term is as high as 66,811 DPMO (i.e. 6.68%). Note that a process operating at a 4.5 Sigma Level of performance is expected to experience 1350 DPMO, which coincides with the one-sided defect rate for the hypothetical process whose specification limits exactly coincide with the three sigma limits.
In this way, the Sigma Level of a process tells us where the specification limits fall relative to the process distribution, assuming that the process may shift as much as 1.5 Sigma over the long term. The 3.4 DPMO quoted for 6 Sigma processes reflects a z-value (in a table of the standard Normal distribution) of 4.5, since the 1.5 sigma shift prevents the process from experiencing its best estimated performance of 2 parts per BILLION defects (a z-value of 6).

Learn more about the Lean Six Sigma principles and tools for process excellence in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, in his online Lean Six Sigma DMAIC short course ($249), or his online Green Belt certification course ($499).