In May 1924, Walter Shewhart wrote a memo that contained the first example of a process behavior chart (i.e., a “control chart”). It was a chart for individual values that would be known today as a p-chart. Shewhart’s insight was that “three-sigma” limits will filter out virtually all of the routine variation so that points outside these limits are likely to represent process changes.
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Following Aristotle’s lead, Shewhart realized that by studying the process changes we can discover their causes and gain insight on how to operate the process more consistently. To recognize the anniversary of Shewhart’s memo, this article outlines some of the history of the process behavior chart.
While Shewhart’s original chart was for proportions, he went on to develop process behavior charts for measurements by using a series of subgroups containing two or more original measurements. These charts plotted an average and dispersion statistic for each subgroup and placed limits of each of these running records. In practice, this quickly became the average and range chart we know today.
The fundamental concept that makes this chart work is the notion of rational subgrouping. According to Shewhart, when we place two values together in the same subgroup we’re making a “judgment” that these two values were collected under “essentially the same conditions.” Thus, the objective of rational subgrouping is to end up with homogeneous subgroups.
As long as we organize the data so the subgroups are internally homogeneous, the average chart will allow us to detect changes that occur between the subgroups, while the range chart will check for consistency within the subgroups. This need for homogeneity within the subgroups will favor the use of smaller subgroup sizes. Regarding this, Shewhart wrote:
“Obviously, if the cause system is changing, the sample [subgroup] size should be as small as possible so that the averages of samples do not mask the changes. In fact, single observations would be the most sensitive to such changes. Why then do we not use a sample size of unity?”1
Shewhart’s answer to this question had two parts: First, with single values we can’t compute a within-subgroup range; and second, using the global standard deviation statistic is unsatisfactory when the cause system is changing. Shewhart then observed that the sensitivity of an average and range chart will increase as the subgroup size decreases until a point is reached where the subgroups are internally homogeneous. In consequence, in the absence of any a priori information about how to organize the data into subgroups, “there would be some advantage in reducing the subgroup size to unity.”
Thus, from the very beginning, the philosophy behind the process behavior chart has been pushing us in the direction of the ‘chart for individual values.’
Thus, from the very beginning, the philosophy behind the process behavior chart has been pushing us in the direction of the “chart for individual values.” Shewhart understood the practical effect of using individual values, but he had a dilemma about how to compute effective limits. Today we resolve this by using the two-point moving ranges, or as they were originally known, the successive differences.
Successive differences
The method of successive differences effectively creates a series of moving subgroups of size two, and computes the ranges for each of these moving subgroups. Thus, with k original values we obtain (k–1) two-point moving ranges. When we average these moving ranges and divide by d2 = 1.128 (the bias correction factor for ranges of subgroups of size two), we obtain an estimate of the standard deviation parameter for the distribution of the original values. (The justification for this use of d2 comes from A. R. Kamat, who was a student of H. O. Hartley.2)
The method of successive differences first appears in the statistical literature in a 1941 paper by John von Neumann of the Institute for Advanced Study at Princeton and three researchers from Aberdeen Proving Ground.3 This paper cites E. Vallier4 as the first to use successive differences for estimating dispersion in 1894, and attributes the first use of the average moving range to C. Crane and K. Becker.5 Since these sources come from ballistics, the idea of using successive differences to characterize dispersion appears to have its origins at the end of the 19th century in determining the range of field artillery.
While von Neumann focused on a statistic known as the mean square successive difference (MSSD), it turns out there is no practical advantage in working with the MSSD rather than the average moving range. When using only two values to compute a measure of dispersion, the standard deviation statistic, s, is simply the range of the two values, R, divided by the square root of 2. As a result of this equivalency, both the MSSD and the average moving range are equally efficient when used with a predictable process.
However, because the MSSD computation squares the differences before averaging them, the MSSD is more easily inflated by extreme range values than is the average moving range. Since extreme range values are to be expected when the process is changing, the average moving range is the preferred statistic for computing robust limits for the XmR chart.
The history of the XmR chart
The XmR chart is attributed to W. J. Jennett, who is said to have created this technique while working with the MO Valve Co. in England in 1942. When we look back at Jennett’s work, we find that in 1942 he co-authored Quality Control Charts.6 However, in this book there’s no mention of the XmR chart. This would suggest that this chart wasn’t around when the book was prepared (in 1941 and earlier). Based on this, it seems likely that Jennett got the idea of using the method of successive differences with individual values from the 1941 von Neumann paper.
Since extreme range values are to be expected when the process is changing, the average moving range is the preferred statistic for computing robust limits for the XmR chart.
The first mention of the XmR chart that I have found in the literature is in L. H. C. Tippett’s 1950 book, where he described the XmR chart using two pages of text and two examples.7 Tippett was the head statistician for the British Cotton Industry Research Association, and in 1948 he was invited to give a series of lectures at Massachusetts Institute of Technology. These lectures were then edited into book form by Shewhart, and the result was Technological Applications of Statistics.
The second mention of the XmR chart comes in 1951 when the American Society for Testing and Materials, Committee E-11 on Quality Control of Materials, issued Version C of its Special Technical Publication 15.8 Here the XmR chart formulas are presented in a couple of pages of text and example. In the past, Shewhart had been instrumental in getting this committee to include the “Control Chart Method of Analysis and Presentation of Data” in Version B of this booklet, and it’s not hard to imagine that he was influential in getting the XmR chart included in Version C. Thus, Shewhart is linked to the first two mentions of the XmR in print, both as an editor and through his friends and associates.
The third mention of the XmR chart is in 1953, in a 10-page article by Joan Keen and Denys Page of the Research Laboratories of the General Electric Co. in Wembley, England.9 In addition to crediting Jennett with creating the technique, Keen and Page present five examples of the XmR chart from what they describe as its extensive use at GE over the preceding decade. (Professor H. O. Hartley appears to have encouraged Keen and Page to write this article to accompany Kamat’s treatment of the theoretical foundation of the moving range technique.)
The next notable mention of the XmR chart is found in the 1956 Western Electric Statistical Quality Control Handbook.10 There, we have about three pages of text and one complete example. When combined, these first four mentions of the XmR chart cover about 17 pages of text and include nine examples. So, there’s substantial evidence that this technique was widely known and used within the first decade following its creation. It was promoted by both Shewhart and Hartley.
However, over the next 30 years, very little was added to this body of material. Some SQC textbooks would not mention the XmR chart at all. Among those that did mention the XmR chart, only about half included an example, and several authors were uncomfortable with this technique. Among this latter group, some felt that the XmR chart would be insensitive to real signals, while others felt that it would be prone to having too many false alarms. These contradictory notions about how the technique might work in practice reveal a widespread lack of familiarity with the XmR chart.
Out of obscurity
So the XmR chart had been around since 1942. While it had been promoted by prominent statisticians (Hartley and Shewhart), and while it was used in industry in the 1940s and ’50s, it appears to have dropped out of use shortly thereafter. When I came along in the ’70s, the XmR chart was little known and rarely used.
Starting in 1982, when I left the university and began full-time consulting and instruction in industry, I did a lot of work with specialty chemical operations. With all sorts of one-at-a-time data, these clients found the XmR chart extremely useful. As I worked with them, I saw the power and utility of the XmR chart over and over again. One of my chemical plant clients managed to win the Ford Q1 award on the first audit. (Up to that time, everyone had failed Ford’s first audit.) My client won the award simply because the XmR charts provided documented answers for virtually every process-related question asked by the auditors.
With all sorts of one-at-a-time data, these clients found the XmR chart extremely useful.
At dinner one evening in 1985, W. Edwards Deming asked me about the XmR chart. He had seen one used by one of my students, and he hadn’t come across this technique before. As we discussed the technique, I explained something of its history and rationale. After looking at some of his own data in this format, and following another discussion, Deming seemed to be satisfied with the role of the XmR chart—he could see the great variety of ways it could be applied. At this same time, David Chambers and I were writing our book that came out in 1986, Understanding Statistical Process Control.11 In it, we included five examples of the XmR chart and 14 pages of text on this topic.
Then, in 1989, Dow Chemical asked me to develop a special module on the managerial uses of the XmR chart for the company’s internal training program. Using the experiences of my clients, I produced that module, and then refined that material into the little book Understanding Variation—the Key to Managing Chaos, which came out in 1993.12 This was the first book dedicated to the XmR chart. Its 32 examples and case histories, and 136 pages of text, easily exceeded all the combined material that had previously been written about this technique. By focusing on managerial types of data, it also expanded the process behavior chart beyond the realm of manufacturing. This book was excerpted in Quality Digest13 and has proven to be one of the most popular books on data analysis ever written. Several companies have credited this little book with turning their operations around.
Today, the XmR chart is an integral part of almost every SPC software package. Yet having access to the technique doesn’t guarantee that it’s always used appropriately. The following discusses some of the issues and questions surrounding the use of the XmR chart.
Rational sampling
In order for the XmR chart to work as intended, two things need to happen. The first of these is that successive values must be logically comparable. The second is that the moving ranges need to capture the routine variation of the underlying process.
A time series that mixes apples and oranges together will not satisfy the two criteria above. You have to organize your data so you are dealing with all apples or all oranges.
You have to exercise some judgment about the relationship between your sample frequency and the way your process operates.
A time series with very short time periods might not allow the successive differences to capture all of the routine variation present in the process. You have to exercise some judgment about the relationship between your sample frequency and the way your process operates.
At the other extreme, if your sample frequency is too low and the process changes in between every one or two observations, then the moving ranges will be inflated by these changes, and the XmR chart will not work as intended. The sample frequency needs to be high enough to result in a situation where most of the moving ranges represent routine variation. (This means that if you’re using monthly values on an XmR chart, you’re assuming that changes occur no more than once or twice a year.)
The name given to this art of getting the right frequency is rational sampling.14 Just as the average and range chart needs logically homogeneous subgroups to work right, the XmR chart needs for most of the moving subgroups to be homogeneous for the moving ranges to capture the routine process variation.
Noise-filled data
Highly aggregated measures will tend to have a lot of noise. (An example would be companywide report-card metrics.) When such measures are placed on an XmR chart, they will tend to show a predictable process. However, the limits will often be so wide that no manager will be pleased with the uncertainty defined by the limits. (Heads will roll, or promotions will occur, long before you reach one limit or the other.)
Here the solution is to disaggregate the time series into its component parts.
Here the solution is to disaggregate the time series into its component parts. As the data become more specific and closer to real time, they become more useful in that they allow you to discover the points where your processes are changing.
So, while charting the highly aggregated report-card measures may present the big picture and make processes look predictable, disaggregating those same measures is essential for learning about your processes and operations. For an example of how this works, see my article in Quality Digest, “Process Behavior Charts as Report Cards.”15
What is the probability model?
“Don’t we need to know if the data are normally distributed before we can use an XmR chart?”
No, it simply doesn’t matter. Countless statisticians and others have foundered on the fallacious idea that the data have to be normally distributed in order for the chart to work.
Regardless of the shape of the histogram, the generic, three-sigma limits of a process behavior chart will filter out approximately 99% or more of the routine variation. Because of this conservative filtration of the routine variation, any point that falls outside the limits may be considered as a potential signal of a process change. To understand this, consider the summary graphic of reference16 given in Figure 1. There we have 19 different probability models covering essentially the full range of gamma models, Weibull models, and lognormal models. Even with these extreme models, the three-sigma limits cover at least 98% of the area in every case.
However, probability models don’t generate our data. Processes do. And in this world, all processes are subject to change. The process behavior chart doesn’t assume any specific probability model for your data, but rather checks to see if your data display enough consistency to make the notion of a probability model make sense. The charts ask the question, “Has a change occurred?” without making reference to any specific probability model.
‘...measurements of phenomena in both social and natural science for the most part obey neither deterministic nor statistical laws until assignable causes of variability have been found and removed.’
—Walter A. Shewhart
As Shewhart wrote in 1943, “Classical statistics start with the assumption that a statistical universe exists, whereas [SPC] starts with the assumption that a statistical universe does not exist....” and also, “...measurements of phenomena in both social and natural science for the most part obey neither deterministic nor statistical laws until assignable causes of variability have been found and removed.”17
When the process is changing, no single probability model can be used to describe the data produced by the process.
What about false alarms?
“But the false alarm rate will change with different probability models.”
Yes, different probability models will allow you to compute different false alarm rates. This is what we should expect with generic, fixed-width limits. For the extreme spectrum of probability models shown in Figure 1, the false alarm rates vary from 1/1,000 to 22/1,000. But these theoretical calculations are simply the sleight of hand by which statisticians distract themselves (and you) from answering the right question.
While a probability model may be used to describe a data set, the process behavior chart isn’t concerned with fitting a model to the process, but rather the opposite. A process behavior chart seeks to fit the process to a model. It seeks to characterize the past process behavior as belonging to one of two classes, either predictable or unpredictable. It asks if the process fits a very broad model of being “predictable within limits.”
When a process is being operated unpredictably, there will be signals of process changes. In this case, having one or two false alarms per hundred points is of no real consequence because the number of signals will usually greatly outnumber the number of false alarms. Here, the story told by the chart will not depend on whether the false alarm rate is 1/1,000 or 2/100.
When a process is reasonably predictable, the routine variation will be common cause variation. This will be the result of many different cause-and-effect relationships where no one cause-and-effect relationship is dominant.18 Under these conditions, Pierre-Simon Laplace’s central limit theorem guarantees that the resulting histogram will tend toward normality, and as this happens the false alarm rate will tend to drop well below 1/100.
So, if the process is unpredictable, the false alarm rate isn’t important. And if it is indeed predictable, then we don’t need to worry about the shape of the histogram because the generic, three-sigma limits will naturally result in a very low false alarm rate.
The Swiss army knife of process behavior charts
By making no presuppositions about your data, the XmR treats them in a completely empirical manner. If you’re not certain about how to organize your data into subgroups, you can start with subgroups of size one—it’s hard to mess up the subgrouping on an XmR chart. As long as you pay attention to the two requirements of comparing apples to apples and allowing the moving ranges to capture the routine process variation, the XmR chart will separate the potential signals from the probable noise. The simplicity of the structure of the XmR chart makes it easy to explain to others. In the words of one supervisor, “Nothing can hide on that XmR chart—every point has to sink or swim on its own.”
So, while it’s possible to put your data on an XmR chart in such a way that the technique will not work, with minimal thought and some process knowledge, you can organize your data so the XmR chart will work as intended. The flexibility of the XmR chart makes it the Swiss army knife of process behavior charts. So ignore the distractions; pay attention to rational sampling; put your data on an XmR chart; and start learning how to improve your quality, your productivity, and your competitive position!
References
1. Shewhart, W. Economic Control of Quality of Manufactured Product. Martino Fine Books, 1931; 2015 reprint, p. 314.
2. Kamat, A.R. “On the Mean Successive Difference and Its Ratio to the Root Mean Square.” Biometrika, vol. 40, 1953, pp. 116–127.
3. Von Neumann, J.; Kent, R.H.; Bellinson, H.R.; Hart, H.R. “The Mean Square Successive Difference.” Annals of Mathematical Statistics, vol. 12, 1941, pp. 153–162.
4. Vallier, E. Balistique Experimentale, 1894.
5. Crane, C.; Becker, K. Exterior Ballistics; translated from German in 1919.
6. Dudding, B.P.; Jennett, W.J. Quality Control Charts. British Standards Institution, 1942.
7. Tippett, L.H.C. Technological Applications of Statistics. John Wiley & Sons, 1950.
8. ASTM Manual on Quality Control of Materials. Special Technical Publication 15-C, 1951.
9. Keen, J.; Page, D. “Estimating Variability from the Differences Between Successive Readings.” Applied Statistics, vol. 2, 1953, pp. 13–23.
10. Statistical Quality Control Handbook. Western Electric, 1956.
11. Wheeler, D.J.; Chambers, D.S. Understanding Statistical Process Control. SPC Press, 1986.
12. Wheeler, D.J. Understanding Variation: The Key to Managing Chaos. SPC Press, 1993.
13. Wheeler, D.J. “Book Excerpt: Understanding Variation.” Quality Digest, vol. 13, Aug., 1993, pp. 22–28.
14. Wheeler, D.J. “Rational Sampling.” Quality Digest, July 1, 2015.
15. Wheeler, D.J. “Process Behavior Charts as Report Cards.” Quality Digest, June 6, 2016.
16. Wheeler, D.J. “Properties of Probability Models: Part Three.” Quality Digest, Oct. 5, 2015.
17. Shewhart, W. “Statistical Control in Applied Science.” Transactions of the ASME, April 1943, pp. 222–225.
18. Wheeler, D.J. “Two Routes to Process Improvement.” Quality Digest, May 5–6, 2010.
Comments
The article, "A History of the Chart for Individual Values."
This takes me back to my days at Dow Corning when I very quickly learned that the Individuals Moving Range chart could be applied to everything.
Thank you for wonderful article, and for the stimulus for me to bounce back to some days filled with fond memories!
Keith Wagoner
Reference #12
If you are in a management position and haven't read Understanding Variation: The Key to Managing Chaos Second Edition , SPCPress 2000, You're missing a real eye opener/
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