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Quality Insider

Published: Wednesday, September 7, 2011 - 09:56

The simplicity of the process behavior chart can be deceptive. This is because the simplicity of the charts is based on a completely different concept of data analysis than that which is used for the analysis of experimental data. When someone does not understand the conceptual basis for process behavior charts, they are likely to view the simplicity of the charts as something that needs to be fixed. Out of these urges to fix the charts all kinds of myths have sprung up, resulting in various levels of complexity and obstacles to the use of one of the most powerful analysis techniques ever invented. The purpose of this article is to help you avoid this complexity.

In discussing this myth some historical background may be helpful. Walter A. Shewhart published his book, *Economic Control of Quality of Manufactured Product,* in 1931. When the British statistician E. S. Pearson read Shewhart’s book, he immediately felt that there were gaps in Shewhart’s approach, and so he set out to fill in these perceived gaps. The result was Pearson’s book entitled, *The Application of Statistical Methods to Industrial Standardization and Quality Control* (British Standards Institution, 1935). In this book Pearson wrote on page 34: “Statistical methods and tables are available to test whether the assumption is justified that the variation in a certain measured characteristic may be represented by the Normal curve.”

After reading Pearson’s book, Shewhart gave a series of lectures that W. Edwards Deming edited into Shewhart’s 1939 book, *Statistical Method from the Viewpoint of Quality Control*. In choosing this title Shewhart effectively reversed Pearson’s title to emphasize that his approach solved a real problem rather than being a collection of techniques looking for an application. On page 54 of this second book Shewhart wrote: *“We are not concerned with the functional form of the universe, but merely with the assumption that a universe exists.”* Here Shewhart went to the heart of the matter. While Pearson essentially assumed that the use of a probability model would always be justified, Shewhart created a technique to examine this assumption. The question addressed by a process behavior chart is more basic than “What is the shape of the histogram?” or “What is the probability model?” It has to do with whether we can meaningfully use any probability model with our data.

Shewhart then went on to note that having a symmetric, bell-shaped histogram is neither a prerequisite for the use of a process behavior chart, nor is it a consequence of having a predictable process. Figure 1 shows Shewhart’s figure 9 from the 1931 book. He characterized these data as “at least approximately [in] a state of control.” This skewed histogram is certainly not one that anyone would claim to be “normally distributed.” So, while Shewhart had thoroughly examined this topic in his 1931 book, his approach was so different from traditional statistical thinking that Pearson and countless others (including this author on his first reading) completely missed this crucial point.

** **

To begin to understand how a process behavior chart can be used with all sorts of data, we need to begin with a simple equation from page 275 of Shewhart’s 1931 book:

Shewhart described two completely different approaches in this equation. The first of these approaches I call the statistical approach since it describes how we approach statistical inference:

1. Choose an appropriate probability model *f(x)* to use;

2. Choose some small risk of a false alarm ( 1 – *P* ) to use;

3. Find the exact critical values *A* and *B* for the selected model that correspond to this risk of a false alarm;

4. Then use these critical values in your analysis.

While this approach makes sense when working with *functions* of the data (i.e., statistics) for which we know the appropriate probability model, it encounters a huge problem when it is applied to the original data. As Shewhart pointed out, we will *never* have enough data to uniquely identify a specific probability model for the original data. In the mathematical sense all probability models are limiting functions for infinite sequences of random variables. This means that they can never be said to apply to any finite portion of that sequence. This is why any assumption of a probability model for the original data is just that—an assumption that cannot be verified in practice. (While lack-of-fit tests will sometimes allow us to falsify this assumption, they can never verify an assumed probability model.)

So what are we to do when we try to analyze data? Shewhart suggested a completely different approach to the equation above. He started by selecting some generic critical values *A* and *B* for which the risk of a false alarm, ( 1 – *P* ) *will be reasonably small regardless* of what probability model *f(x)* we might choose. This approach changed what is fixed and what is allowed to vary. With the statistical approach the risk of a false alarm is fixed, and the critical values vary to match the specific probability model. With Shewhart’s approach it is the critical values that are fixed (the three-sigma limits) and the risk of a false alarm that is allowed to vary. This complete reversal of the statistical approach is what makes Shewhart’s approach so hard for those with statistical training to understand.

Once you see the difference in these two approaches, you can begin to see why Pearson and others have been concerned with the probability model *f(x)*, why they have sought to maintain a fixed alpha level ( 1 – *P* ), and why they have been obsessed with the computation of exact values for *A* and *B*. And more recently, you can see how others have become obsessed with transforming the data prior to placing them on a process behavior chart. Their presuppositions prevent them from understanding how Shewhart’s choice of three-sigma limits is completely independent of the choice of a probability model. In fact, to their way of thinking, you cannot even get started without a probability model. Hence, as people keep misunderstanding the basis for process behavior charts, they continue to recreate Myth One.

The central limit theorem was published by Laplace in 1810. This fundamental theorem shows how, regardless of the shape of the histogram of the original data, the histograms of subgroup averages will tend to have a “normal” shape as the subgroup size gets larger. This is illustrated in figure 2, where the histograms for 1000 subgroup averages are shown for each of three different subgroup sizes for data obtained from two completely different sets of original data. There we see that even though the histograms for the individual values differ, the histograms for the subgroup averages tend to look more alike and become more bell-shaped as the subgroup size increases.

Many statistical techniques that are based on averages utilize the central limit theorem. While we may not know what the histogram for the original data looks like, we can be reasonably sure that the histogram of the subgroup averages may be approximated by a normal distribution. From this point we can then use the statistical approach outlined in the preceding section to carry out our analysis using the subgroup averages.

However, while we have a central limit theorem for subgroup averages, there is no central limit theorem for subgroup ranges. This is illustrated in figure 3 where we see the histograms of the subgroup ranges obtained from two different sets of original data. Each histogram shows 1000 subgroup ranges found using each of three subgroup sizes. As the subgroup size increases the histograms for the subgroup ranges become more dissimilar and do not even begin to look bell-shaped.

Therefore, Myth Two has no basis in reality. If the central limit theorem was the foundation for process behavior charts, then the range chart would not work.

Rather, as we saw in the preceding section, Shewhart chose three-sigma limits to use with the process behavior chart simply because when the data are homogeneous, these limits will bracket virtually all of the histogram regardless of the shape of that histogram. Three-sigma limits are shown on each of the 16 histograms in figures 2 and 3. There they bracket better than 98 percent of each histogram, leaving less than a 2-percent chance of a false alarm in each case. In practice, as long as ( 1 – *P* ) is known to be small, you do not need to know the exact risk of a false alarm. This means that when you find a point outside the limits of a process behavior chart, the odds are very good that the underlying process has changed and you will be justified in taking action. Three-sigma limits provide you with a suitably conservative analysis without requiring a lot of preliminary work. It is this conservative nature of three-sigma limits that eliminates the need to appeal to the central limit theorem to justify the process behavior chart.

Undoubtedly, Myth Two has been one of the greatest barriers to the use of process behavior charts with management data and process-industry data. Whenever data are obtained one-value-per-time-period it will be logical to use subgroups of size one. However, if you believe Myth Two, you will feel compelled to average something in order to invoke the blessing of the central limit theorem, and the rationality of your data analysis will be sacrificed to superstition. The conservative nature of three-sigma limits allows you to use the chart for individual values with all sorts of original data without reference to the shape of the histogram.

Again we have an artificial barrier to the use of a process behavior chart, which ignores both the nature of real data and the robustness of the process behavior chart technique. Virtually all data coming from a production process will display some amount of autocorrelation.

Autocorrelation is simply a measure of the correlation between a time series and itself. A positive autocorrelation (lag one) simply means that the data display two characteristics:

1. Successive values are generally quite similar.

2. Values that are far apart can be quite dissimilar.

These two properties mean that when the data have a large positive autocorrelation, the underlying process will be changing. To illustrate this property I will use the data from Table 2, page 20 of Shewhart’s 1931 book. These data are the measured resistances of insulation material. These data have an autocorrelation of 0.549, which is detectably different from zero (also known as significantly different from zero). While Shewhart organized these data into 51 subgroups of size four and placed them on an average chart, it could be argued that this subgrouping obscures the effects of the autocorrelation upon the chart. To avoid this problem I have placed these 204 data on an XmR chart in figure 4.

Shewhart found eight averages outside his limits. We find 14 individual values and seven moving ranges outside our limits. So both Shewhart’s average chart and our XmR chart tell the same story. This process was not being operated predictably.

As they found the assignable causes and took steps to remove their effects from this process they collected some new data. These data, shown in figure 5, show no evidence of unpredictable behavior. Notice that the new limits are only 60 percent as wide as the original limits. By removing the assignable causes of exceptional variation, they not only got rid of the process upsets and the extreme values, but they also removed a substantial amount of process variation. The autocorrelation for the data in figure 5 is 0.091, which is not detectably different from zero.

This example illustrates an important point. Whenever the data have a substantial autocorrelation, the underlying process will be changing, and vice-versa; when the process is moving around, the data will tend to have an autocorrelation that is detectably different from zero. Thus, autocorrelation is simply one way that the data have of revealing that the underlying process is changing. On the other hand, when the process is operated predictably, the data are unlikely to possess a substantial autocorrelation.

Remember that the purpose of analysis is insight rather than numbers. The process behavior chart is not concerned with creating a model for the data, or whether the data fit a specific model, but rather with using data for making decisions in the real world. To insist that the data be independent is to add something to Shewhart’s work that Shewhart was careful to avoid. This example from Shewhart’s first book illustrates that process behavior charts have worked with autocorrelated data from the very beginning. Do not let those who do not understand this point keep you from placing your data on a chart because the values might not be independent.

Although a complete treatment of the effects of autocorrelation is beyond the scope of this article, the following observation is in order. While it is true that when the autocorrelation gets close to +1.00 or –1.00 the autocorrelation can have an impact upon the computation of the limits, such autocorrelations will also simultaneously create running records that are easy to interpret at face value. This increased interpretability of the running record will usually provide the insight needed for process improvement and further computations become unnecessary.

I first encountered this myth when I was refereeing a paper written by a professor of statistics at a land-grant university in the South, which goes to prove my point that even an extensive knowledge of statistics does not guarantee that you will understand Shewhart.

I suspect that the origin of Myth Four is a failure to appreciate that there are correct and incorrect ways of computing the limits for a process behavior chart. (See my January and February *Quality Digest* columns from 2010 for more on this topic.) The most common of the incorrect ways of computing limits consists of using three-standard-deviation limits rather than three-sigma limits. While this approach was identified as incorrect on page 302 of Shewhart’s 1931 book, it is found in virtually every piece of software available today. While three-standard-deviation limits will mimic three-sigma limits whenever the process is operated predictably, they will be severely inflated when the process is being operated unpredictably. Thus, when someone is using the incorrect way of computing the limits, they might come to believe Myth Four.

Of course, as soon as you believe Myth Four, you will begin to look for a way to remedy this perceived defect in the technique. Among the absurdities which have been perpetrated in the name of Myth Four are censoring of the data prior to placing them on the chart (removing the outliers), and the use of two-standard-deviation limits. (As Henry Neave observed in a letter to the Royal Statistical Society, calculating the limits incorrectly and then using the wrong multiplier is an example of how two wrongs still do not make one right.) Needless to say that these, and all other associated manipulations, are unnecessary. The express purpose of the process behavior chart is to detect when a process is changing, and to do this we have to be able to get good limits from bad data.

Terry, one of my students, had just completed the class and was looking at the archival data he had for his cooling water system. He organized these data into daily subgroups of size five and plotted the averages and ranges for the past 24 days to get the graph shown in figure 6.

Based on what he saw in figure 6, Terry decided to use the first half of the data to compute the limits for this chart. When he did this, he got the limits shown in figure 7.

The two high points coincided with the short week prior to the Christmas break, and the drop to the lower points coincided with the January start-up. Based on this chart, Terry was able to explain why they were spending more than $2 million a year on scrap product. With some minor changes they immediately cut the scrap rate by 70 percent, and by the end of the following year they had cut the scrap rate to 10 percent of what it had been by simply operating their processes more consistently.

But why did Terry only use the first 12 days in computing the data? Because he was afraid that the limits would “blow-up” if he used all of the data. Figure 8 shows the chart of figure 7 with two sets of limits. The limits shown as solid blue lines were computed using the data from all 24 days. The red dashed lines show Terry’s limits.

While the limits do change slightly, the story told by the chart remains the same regardless of which set of limits you use. The purpose of a process behavior chart is to tell the story contained within the data and the limits are simply a means to this end.

Thus, as illustrated by figure 8, we can compute good limits using bad data. We do not have to wait until the process is “well-behaved” before we compute our limits. The correct computations are robust. And this is why Myth Four is patent nonsense.

Shewhart’s approach to the analysis of data is profoundly different from the statistical approach. This is why people end up with such confusion when they try to “update” Shewhart by attaching bits and pieces from the statistical approach to what Shewhart has already done. As I showed in my March column, “Three Questions for Success,” Shewhart provided us with an operational definition of how to get the most out of any process. Nothing extra is needed to make process behavior charts work. We do not need to check for normality or transform the data to make them “more normal.” We do not have to use subgrouped data in order to receive the blessing of the central limit theorem before the chart will work. We do not need to examine our data for autocorrelation. And we do not need to wait until our process is “well-behaved” before computing limits. All such “extras” will just mess you up, get in your way, leave you confused, and keep you from using one of the most powerful data analysis techniques ever invented.

## Comments

## Myth 1

Dr. Wheeler,

The last two paragraphs under Myth 1 were very enlightening for me. Thanks for explaining something that I have wondered about for a long time. These two paragraphs were an "A-Ha! moment" for me.

Steve Moore

## AIAG Books

Don't put too much faith in the AIAG manuals. Both the MSA and FMEA books publisheed by AIAG have serious flaws (as Dr. Wheeler has pointed out).

Learn SPC from a master.

Rich DeRoeck

## subgrouping

Dr Wheeler,

If we do not have to use subgrouped data, why is subgrouping used and explained in your books?

## Reply for Jose Arreola

Jose,

We subgroup for many reasons, but some will insist that we have to subgroup the data to invoke the blessing of the central limit theorem. Thus my comment was saying that subgrouping is not mandatory, not that it was not useful in many cases. When producing widgets, where we can choose both the subgroup size and the subgroup frequency, subgrouping is recommended. When working with data that come one number at a time, we usually will want to use subgroups of size one, and here subgrouping can get in the way.

## Fact vs the popular view

Another excellent article. However you can be sure that the masses will not let the truth get in the way of popular opinion. The myths of Six Sigma rule ! There is a strong parallel with the man caused global warming scam. There is not a shred of evidence to support AGW but the masses still follow blindly.

## Thank you so much for an eye-opening article.

We've been taught in school and have been reinforced and practiced in work that to use process behavior charts, the data needs to be normal and that there should be no outliers.

Now I know that:

Hoping that your article will help our organization see the correct way of using process behavior charts.

Presenting this article will be a challenge, but worth the effort.

-Red Anderson

## It depends on the situation

Red,

I have ALWAYS included tests for distributional fit (histogram and chi square test, and quantile-quantile plot) in any process capability study (including control chart preparation) I have ever done, and I recall counseling suppliers whose reports it was my job to review to do the same. First, if it isn't normal, the process capability estimate can be off by orders of magnitude (as measured by nonconforming fraction) especially if you have a gamma distribution with a long upper tail--the end at which the specification limit usually lies. A "Six Sigma" process can in such a case give you 10 or more DPMO and that is without the 1.5 sigma process shift assumed by Motorola.

Second, the false alarm risk can be enormous, and I am again talking about at least an order of magnitude in extreme cases. The practical implication is that production workers will waste time and eventually lose confidence in SPC. The apparent Weibull distribution in Figure 1, by the way, looks sufficiently bell-shaped that the false alarm risk will not be extreme and the traditional Shewhart chart might be adequate for it. There are in fact practical situations in which, even if the critical to quality characteristic is known to be non-normal, it will behave sufficiently normal so as to pass goodness of fit tests for normality.

From what I have seen of sample ranges, the distribution of R becomes more bell shaped with increasing sample size. The same for the s chart noting that the chi square distribution (for s squared) becomes more bell shaped with increasing degrees of freedom.

It is to be noted that Shewhart developed his methods in the 1920s and 1930s, during which it would have been computationally prohibitive to set exact control limits for non-normal distributions. It is now almost routine to do so with StatGraphics, Minitab, or even spreadsheet functions. Therefore, it seems reasonable to do it that way; fit the actual distribution, set control limits for the desired false alarm risks, and calculate the process performance index that corresponds to the nonconforming fraction (an approach now approved by AIAG's SPC manual).

## Reply to William Levinson

Bill, you are completely wrong.

## AIAG on non-normal charts

Donald,

AIAG's Statistical Process Control manual (2nd ed, p. 113) agrees with you that Shewhart did not develop his charts under the assumption of normality, and that the charts can be used for all processes as you say. It also agrees with me that "However, as the process distribution deviates from normality, the sensitivity to change decreases, and the risk associated with the Type I error increases."

The manual also says you can use standard Shewhart control charts (as you recommend) with appropriate sample sizes, OR adjust the limits to reflect the non-normal form, OR use a transformation, OR use control limits based on the native non-normal form (which is what I recommend). "Appropriate sample sizes" seems to suggest reliance on the Central Limit Theorem, which certainly mitigates the effects of non-normality.

Even if the traditional Shewhart chart is adequate for control purposes in the shop (as seems likely for Figure 1), though, one must still deal with the process capability estimate. According to pages 140-143 of the AIAG reference, you can compute the nonconforming fraction and the corresponding normal-equivalent performance index (e.g. 1 ppb above the upper specification limit => PPU = 2.0). Another is to compute Pp = (USL-LSL)/(Q(0.99865)-Q(0.00135)) where Q refers to the quantile of the distribution. The latter, also known as the ISO method, is attractive because it converts directly to Pp = (USL-LSL)/(6 standard deviations) if the distribution is normal.

Use of either of these, however, requires us to fit the non-normal distribution to the process data. Since we HAVE to do this to provide a meaningful performance index (I say "performance" rather than "capability" because the maximum likelihood estimation methods rely on the body of the data instead of subgroups), we may as well then use the fitted distribution to set control limits with known false alarm risks and average run lengths.

## Thank you!

Dear Dr. Wheeler,

Thank you for your column explaining the myths away. I was just discussing how important it was to understand an idea's historical context to appreciate its intent. It was a pleasant coincidence to see you do that in your explanation. With distance in time, it is easy to lose the operational definition of an idea and inadvertently corrupt it to fit a particular perspective.

I will share your post with others to reconnect them with Dr. Shewhart's intent.

Regards,

Shrikant Kalegaonkar

twitter: shrikale