The ±1.5 Sigma Drift

The ±1.5s drift is the drift of a process mean, which is assumed to occur in all processes. If a product is manufactured to a target of 100 mm using a process capable of delivering s = 1 mm performance, over time a ±1.5s drift may cause the long term process mean to range from 98.5 to 101.5 mm. This could be of significance to customers.

The ±1.5s shift was introduced by Mikel Harry. Harry referred to a paper about tolerancing, the overall error in an assembly is affected by the errors in components, written in 1975 by Evans, "Statistical Tolerancing: The State of the Art. Part 3. Shifts and Drifts". Evans refers to a paper by Bender in 1962, "Benderizing Tolerances – A Simple Practical Probability Method for Handling Tolerances for Limit Stack Ups". He looked at the classical situation with a stack of disks and how the overall error in the size of the stack, relates to errors in the individual disks. Based on "probability, approximations and experience", Bender suggests:

A run chart depicting a +1.5s drift in a 6s process. USL and LSL are the upper and lower specification limits and UNL and LNL are the upper and lower natural tolerance limits.

Harry then took this a step further. Supposing that there is a process in which 5 samples are taken every half hour and plotted on a control chart, Harry considered the "instantaneous" initial 5 samples as being "short term" (Harry's n=5) and the samples throughout the day as being "long term" (Harry's g=50 points). Due to the random variation in the first 5 points, the mean of the initial sample is different from the overall mean. Harry derived a relationship between the short term and long term capability, using the equation above, to produce a capability shift or "Z shift" of 1.5. Over time, the original meaning of "short term" and "long term" has been changed to result in "long term" drifting means.

Harry has clung tenaciously to the "1.5" but over the years, its derivation has been modified. In a recent note from Harry, "We employed the value of 1.5 since no other empirical information was available at the time of reporting." In other words, 1.5 has now become an empirical rather than theoretical value. Harry further softened this by stating "... the 1.5 constant would not be needed as an approximation". Interestingly, 1.5s is exactly one half of the commonly accepted natural tolerance limits of 3s.

Despite this, industry is resigned to the belief that it is impossible to keep processes on target and that process means will inevitably drift by ±1.5s. In other words, if a process has a target value of 0.0, specification limits at 6s, and natural tolerance limits of ±3s, over the long term the mean may drift to +1.5 (or -1.5).

In truth, any process where the mean changes by 1.5s, or any other statistically significant amount, is not in statistical control. Such a change can often be detected by a trend on a control chart. A process that is not in control is not predictable. It may begin to produce defects, no matter where specification limits have been set.