Sometimes when authors try to make a technical concept more understandable, it’s simplified but unfortunately, less accurate.
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For example, when the developers of Six Sigma wanted to explain control charts and process capability analysis, they needed to include how the signal can be separated from the noise in time-series data. Instead of falling back on terminology used by Walter A. Shewhart, W. Edwards Deming, Acheson J. Duncan and others, they created new terms to explain subgrouping and the within-and-between subgroup variation. Perhaps they felt the old terminology was too abstract and confusing for practitioners. Apparently they chose to skip over important details like what the difference is between a subgroup and a rational subgroup, and deemed “short-term” and “long-term” as sufficient.
“Everything should be made as simple as possible, but not simpler.”
—Albert Einstein
At first this terminology might seem to be reasonable because process observations taken close together in time will frequently form what is called a rational subgroup because the input factors are not changing significantly within this short-term time window. But what is rational subgrouping and why is this undefined term important? A rational subgroup is a subset of the time-series data set in which the causal factors affecting the variation within the subgroup are homogeneous. That is, the factors causing variation within the subgroup remain the same and are only experiencing random variation. More precisely for a static process (i.e., one that satisfies the Shewhart model , the causal factors, typically the 5Ms and 1E (i.e., manpower, machines, material, methods, measurement, and the environment) are independent and randomly distributed with fixed mean and fixed variance.
Although the phrase “short-term” seems like an easy-to-understand concept, it is, unfortunately, subjective in nature (i.e., how short is short and what does shortness have to do with rationality?) and it’s a far cry from telling the practitioner why rational subgrouping is critically important or how to select a rational subgroup from process data. Instead of an approach that leads the practitioner to the proper understanding and the correct answer, this simplification can lead the practitioner to no real understanding and an incorrect answer for process behavior charts and process capability analysis.
Example 1: Being too “short-term” can lead to the wrong result.
One hundred readings are taken from a high-speed production process where the observations are taken very close together in time.
Question: Is the process in control?
The process data: N = {x1, x2, x3, x4, x5, …
Select the subgroup size n = 1.
Analyze the process with an Individuals and two-point moving range (mR) control chart.
mR = Avg (R{xi, xi+1}) i = 1 to 100.
Now, if the data are autocorrelated (which is a common problem with data from time-series process data taken too close together in time), then the estimate of sigma will be biased low. The control limits and process capability estimates generated from this incorrect sigma will produce a control chart with too many out-of-control alarm signals (false positives) and too large an estimate for the potential process capability.
The data being “short-term” does not assure that you will derive the correct answers from the analysis. You need to understand that the objective of a process behavior chart is the assessment of process stability.
To quote Donald J. Wheeler: “A key aspect of the efficient use of the process behavior chart is making the charts answer the right questions. In order to do this, the way the data are organized into subgroups must be matched with the structure present in the data. This usually means that each subgroup will be selected from some small region of space, or time, or product, in order to assure relatively homogeneous conditions within the subgroup.” This is rational subgrouping and it’s the heart and soul of proper process behavior chart analysis.
Unfortunately selecting a subgroup based on a practitioner’s subjective judgment that the size or length is “short enough” can be a recipe for disaster. On one occasion a highly skilled engineer told me that he had selected the subgroup size of 50 for the data set he was analyzing.
“That might cause a problem,” I told him.
“The formula for computing the control limits works,” he responded.
“Yes, the formulas work, but the underling basis of control chart design might be violated,” I explained. “You could be capturing a mixture of both special and common-cause variation in the subgroup when the goal is to only capture common-cause variation in the rational subgroup used to compute the control limits.”
What’s needed is a procedure for determining if a subgroup is “sufficiently” rational to generate a good estimate of common-cause variation. Below is a proposal for such a procedure.
The rational subgrouping method for Shewhart control charts:
1. Select a subgroup size n (typically this should be a number from 1 to 5, but numbers as high as 10 may be acceptable).
2. Select an ordered subset of observations from the time series population for the proposed rational subgroup.
2. Run an autocorrelation function (ACF) test using various lags on the population data set N.
3. Select the observations for the subgroup where the ACF lag is large enough so that there is no significant autocorrelation between the xi.
4. Next check that the process causal factors (i.e., the 5Ms and 1E) are the same and their performance distribution is static over the study time (i.e., homogenous), which means they’re only experiencing random variation. We check this because changes in any one of the critical causal factors may undermine homogeneity and this can happen if the subgroup sample is too long-term. In addition, to determine if the causal factors are sufficiently homogenous we need to generate a control chart for each input variable, and all of the charts must indicate process stability.
5. If all these conditions are met, then you have a rational subgroup.
The reason for analyzing the process this way is to get an estimate of how good the process could be if everything was operating “perfectly” (i.e., all input factors are experiencing only random variation and the process is said to be “in control”). The control chart is built on this perfect performance model so when future performance data are plotted on the chart in phase II, we can see if the behavior of the process is static or has changed for better or worse.
Example 2: Control chart phase I
A production process with N = 100 measurement observations taken five times per shift on two shifts.
Question: Is the process in control and is there a difference in shift performance?
N = {x1, x2, x3, x4, x5, …
Select the trial subgroup size n = 2.
Generate the ACF for N at lags 1 through 20.
ACF at lag 1 = 0.8, ACF at lag 2 = 0.3, ACF at lag 3 = 0.1, ACF at lag 4 = 0.05.
Since a lag of 3 assures a low level of autocorrelation, then select an “independent” data subset of N called N* = {x1, x4, x7, x10, ….
Form from N* a time-series set of trial rational subgroups {x1, x4}, {x7, x10}, {x13, x16}, … where the first, third, etc. subgroups are from the day shift (i.e., odd numbered) and the even numbered subgroups are from swing shift.
Homogeneity test example: The process data in set N was taken five readings per shift, and within the odd number subgroups of N*, the 5Ms and 1E causal factors are independent and have remained essentially the same. This is a common condition associated with rationality, but it’s not sufficient to confirm it. To confirm it a control chart for each causal factor must be stable because even though the factor is fixed, the behavior of it may not be random. If all the control charts are stable, this establishes that subgrouping is rational. Then use the odd numbered rational subgroups because they are only experiencing common-cause variation to compute trial control limits for the mean and range control chart. Next plot all (i.e., both odd and even) trial subgroup mean and range data on the process behavior chart. If there are no out-of-control signals based on the Western Electric test rules, then conclude that the process is “in control” and that the performance of both shifts are not significantly different.
Comments
The real origins of Sick Sigma's "Long Term"
The origin of the common Six Sigma "long term" expression is explained in detail below. It came from that farcical "long term drift/shift" of +/-1.5 sigma that forms the "six sigma" of Six Sigma. The clear advice is stick to Shewhart Charts.http://www.qualitydigest.com/inside/six-sigma-article/sick-sigma.html"
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The +/-1.5 shift was introduced by Mikel Harry. Where did he get it? Harry refers to a paper written in 1975 by Evans, “Statistical Tolerancing: The State of the Art. Part 3. Shifts and Drifts.” The paper is about tolerancing. That’s how the overall error in an assembly is affected by the errors in components. 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 related to errors in the individual disks. Based on probability, approximations and experience, he asks:
"How is this related to monitoring the myriad processes that people are concerned about?" Very little. Harry then takes things a step further. Imagine a process where five samples are taken every half hour and plotted on a control chart. Harry considered the “instantaneous” initial five samples as being “short term” (Harry’s n=5) and the samples throughout the day as being “long term” (Harry’s g=50 points). Because of random variation in the first five 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 to produce a capability shift or “Z shift” of 1.5. Over time, the original meaning of instantaneous “short term” and the 50-sample point “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, Harry writes, “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. A further softening from Harry: “… the 1.5 constant would not be needed as an approximation.”
My work with Dr. Mikel Harry on 1.5 Sigma Shift
On October of 2013, I had the pleasure of working with Dr. Mikel Harry on his 1.5 Sigma Shift Theory. The paper that I proofed for him took us a long time to bring together. The theory appears valid, and I would welcome discussion from anyone in the community who would like to review and discuss further with me personally.
Please refer to my website Rent-A-Blackbelt.com for contact information - I look forward to speaking with you about this topic.
Best,
Mr. Manny Barriger, CEO
Rent-A-Blackbelt ®
long term vs short term variation
Good evening everyone!
It may be a silly question but i wanted to ask if there is any chance that the short term variation can be larger than the long term variation! Thanks in advance!!
Short term vs long term variation
Surprise, surprise. Short term variation can exceed long term. I have seen this happen several times but it is not real. It can happen in small samples due to rounding.
--John J. Flaig, Ph.D.
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