Introduction - Understanding the Six Sigma Philosophy

Much discussion in recent years has been devoted to the concept of "six sigma" quality. The company most often associated with this philosophy is Motorola, Inc., whose definition of this principle is stated by Harry (1997, p. 3) as follows :

A product is said to have six sigma quality when it exhibits no more than 3.4 npmo at the part and process step levels.

Confusion often exists about the relationship between six sigma and this definition of producing no more than 3.4 nonconformities per million opportunities. From a typical normal distribution table, one may find that the area underneath the normal curve beyond six sigma from the average is 1.248 — 10 ^-9 or .001248 ppm, which is about 1 part per billion. Considering both tails of the process distribution, this would be a total of .002 ppm. This process has the potential capability of fitting two six sigma spreads within the tolerance, or equivalently, having 12 ƒ equal the tolerance.

However, the 3.4 ppm value corresponds to the area under the curve at a distance of only 4.5 sigma from the process average. Why this apparent discrepancy? It is due to the difference between a static and a dynamic process. (The reader is encouraged to review Volume I of this series.)

A STATIC VERSUS A DYNAMIC PROCESS

If a process is static , meaning the process average remains centered at the middle of the tolerance, then approximately .002 ppm will be produced. But under the six sigma concept, the process is considered to be dynamic , implying that over time, the process average will move both higher and lower because of many small changes in material, operators, environmental factors, tools, etc. Most small shifts in the process average will go undetected by the control chart. For an n of 4, there is only a 50 percent chance a 1.5-sigma shift in ¼ is detected by the next subgroup after this change. By the time this next subgroup is collected, it may have returned to its original position. Thus, this process change will never be noticed on the chart, which means that no corrective action will be implemented. However, this movement has caused the actual long-term process variation to increase somewhat because between-subgroup variation is greater than within-subgroup variation. Note that estimates of short- term process variation are not impacted because they are determined only from within-subgroup variation.

Based on studies analyzing the effect of these changes on process variation (Bender, 1962, 1968; Evans, 1970, 1974, 1975a and b; Gilson, 1951), the six sigma principle acknowledges the likelihood of undetected shifts in the process average of up to ±1.5 sigma. Because shifts in the average greater than 1.5 sigma are expected to be caught, and six is assumed not to change, the worst case for the production of nonconforming parts happens when the process average has shifted either the full 1.5 sigma above the middle of the tolerance or the full 1.5 sigma below it. For this worst case, there would be only 4.5 sigma (6 sigma minus 1.5 sigma) remaining between the process average and the nearest specification limit.

This reduced Z value of 4.5 for the dynamic model corresponds to 3.4 ppm. When this size of shift occurs, the Z value for the other specification limit becomes 7.5, which means essentially 0 ppm are outside this limit. Because the process average can shift in only one direction at a time, the maximum number of nonconforming parts produced is 3.4 ppm. Notice that most of the time the average should be closer to the middle of the tolerance, resulting in far fewer than 3.4 ppm actually being manufactured.

To achieve a goal of 3.4 ppm, the process average must be no closer than 4.5 sigma to a specification limit. Assuming the average could drift by as much as 1.5 sigma, potential capability must be at least 6.06 (4.56 plus 1.5 sigma) to compensate for shifts in the process average of up to 1.56, yet still be able to produce the desired quality level. The required 4.56 plus this added buffer of 1.5 sigma create the 6 ƒ requirement, and thereby generate the label "six sigma." (Here it must be noted that the 4.5 shift is allegedly an empirical value for the electronic industry. In the automotive industry, for years the shift has been identified as only 1 sigma ” a shift from a P _pk of 1.33 to a C _pk of 1.67 i.e., from 4 sigma to 5 sigma. The point is that every industry should identify its own shift and use it accordingly . It is unfortunate that the 4.5 shift has become the default value for everything. For a detailed explanation on the difference between P _pk and C _pk , the reader is encouraged to review Volumes I and IV of this series.)

To counter the effect of shifts in ¼ , a buffer of 1.5 standard deviations can be added to other capability goals as well. If no more than 32 ppm are desired outside either specification, the goal would be to have ±4.06 fit within the tolerance, assuming no change in the process average. This target equates to a C _p of 1.33 (4.0/3). Under the static model, this potential capability goal translates into 32 ppm outside each specification when the average is centered at M. But with the inevitable 1.56 drifts in ¼ occurring with the dynamic process model, the average could move as close as 2.56 (4.5 sigma minus 1.5 sigma) to a specification limit before triggering any type of corrective action. This change in centering would cause as many as 6210 ppm to be produced, quite a bit more than the desired maximum of 32 ppm.