Mar 2 2019

# The Math Behind The Process Behavior Chart

Ever since asking Is SPC Obsolete? on this blog almost 6 years ago, multiple sources have told me that the XmR chart is a wonderful and currently useful process behavior chart, universally applicable, a data analysis panacea, requiring no assumption on the structure of the monitored variables. So I dug into it and this what I found.

## What’s an XmR chart?

Here is an example of what these charts look like:

### The X chart and the mR chart

The top chart shows individual measurements over time; the bottom one, a moving range based on the last two values. What makes this more than a plain times series charts — the likes of which are found in as unsophisticated a publication as USA Today — is the red dashed lines marking control limits.

*Misleading time series from USA Today*

## How control limits are set

The method for setting these limits is essential to the tool.

### The Recipe

The limits are multiples of the average of the moving range \overline{mR} . What the literature tells you is that, for the chart of individual values, the limits are at:

Average \pm 2.66\times\overline{mR}

and for the range at

3.27\times\overline{mR}

### Concerns about the formulas

The readers are discouraged from worrying their heads with the provenance of these numbers, and the closest they get to an explanation is a reference to a table where such coefficients are kept, and this table was generated by a higher authority. No further explanation is found in the books I have seen by Douglas Montgomery, J.M. Juran, or Don Wheeler.

The assertion that these numbers are valid regardless of the data’s variation pattern strains credulity. Clearly, they did not come out of thin air. There is a theory behind them, that theory is based on assumptions about the data, and it is necessary for users to know these assumptions so that they can understand the domain of applicability of the technique, with its blurry boundaries.

You may still get some use out of the technique when the conditions are not fully met but you should do so knowingly. As discussed in an earlier comment, chemical engineers commonly apply formulas for perfect gases to gases they know aren’t perfect.

### The math of limit setting

Not finding this information for the XmR chart, I undertook to work it out myself, starting with the model implicit in the SPC literature, that a process variable *X* is the sum of a constant *C* with a white noise *W*. Formally, for the i-th measurement,

where the W_{i} are independent Gaussian (also known as “Normal”) variables with 0 mean and standard deviation σ. Let us consider the differences between consecutive variables X_{i} and X_{i-1}:

X_{i} - X_{i-1} = W_{i} -W_{i-1}

As sums of two independent Gaussian variables with 0 mean and standard deviation σ, they are also Gaussian, with 0 mean and standard deviation \sqrt{2}\times\sigma . The range R_{i} is the absolute value of this difference:

R_{i} = \left |X_{i} - X_{i-1} \right | = \left |W_{i} - W_{i-1} \right |

and follows the Half-Gaussian distribution. The mean \mu_{R} of the range is:

\mu_{R} = \sqrt{\frac{2}{\pi}}\times(\sqrt{2}\sigma) = \frac{2}{\sqrt{\pi}}\times\sigma

and therefore:

\sigma = \frac{\sqrt{\pi}}{2}\times\mu_{R} = 0.8862 \times\mu_{R}

which results in:

3\sigma = 2.66\times\mu_{R}

For the standard deviation \sigma_{R} of the moving range, we have:

\sigma_{R}^{2 }= 2\sigma^{2} -\mu_{R}^{2} = \mu_{R}^{2}\times(\frac{\pi}{2} -1)

This means that, for R, the r\sigma_{R} upper control limit is:

\mu_{R} + 3\sigma_{R} = \mu_{R}\times\left (1 + 3\times\sqrt{\frac{\pi}{2} -1} \right ) = 3.27\times\mu_{R}

This derivation confirms that the published coefficients are based on the assumption that the monitored variable is Gaussian. Of course, it doesn’t *prove* it but it is highly unlikely that any other distribution would produce the exact same coefficients.

## Interpretation

We need to consider the two charts separately.

### The X chart

The X chart is simply a plot of the raw time series and, as long as the model X_{i} = C + W_{i} holds, 99.7% of the points will be within ±3σ of C. This is now commonly described as saying that the “p-value” of checking against these limits is 0.3%.

Generally, the p-value of a sample statistic is the probability that it will be outside its limits when all the data are generated from the reference model. P-values are often used today because they are more general and less arbitrary than levels of significance, and easy to calculate. A 95% level of significance means p=.05 and 99%, p=.01. Running multiple tests against the same data until you find one that gives you a “significant” difference is called p-hacking.

### The mR chart

The p-value of the mR chart, however, is higher, meaning that it is more prone to false alarms. In terms of \sigma, the mR chart’s upper control limit is

3.27\times\mu_{R} = 3.69\sigma

and the standard deviation of the differences W_{i} - W_{i-1} is

\sigma_{D} = \sqrt{2}\times\sigma

If we plug these numbers into the formula for the cumulative distribution function of the half-gaussian, we get:

p = 1- erf(\frac{3.69}{2}) = 1\%

As can be easily verified with simulations, this yields an average of 1 false alarm for every 100 points, which is three times more than the X chart. If the process is unstable, there are so many genuine alarms for the engineering team to investigate that false alarms won’t be a problem.

If, on the other hand, it is so stable that it never goes out of control, the chart will still generate alarms and they will all be false. After two or three cases of assembling a task force to chase non-existent assignable causes, management will lose confidence in the chart.

## Don Wheeler’s reasons for plotting mR charts

XmR chart expert Don Wheeler offers the following, not-fully compelling reasons to plot mR charts:

- “The Moving Range Chart will, on occasion, provide new information in addition to reinforcing the message of the X Chart.”
- “Thus, the mR Chart is the secret handshake of those who know the correct way of computing limits for an X Chart. Omit it and your readers cannot be sure that you are a member of the club.”
- “The mR Chart allows you and your audience to check for the problem of chunky data. This is a problem that occurs when the data have been rounded to the point that the variation is lost in the round-off.”

### New information?

The mR chart is indeed a summary of the X chart and, as such, may provide new information. The questions are whether it is worth the trouble to maintain it and whether there are no other tools from time series analysis to get the same information more easily.

### Club membership

The point of visualizing and analyzing quality characteristics is to troubleshoot a process, not to secure the approval of others or prove membership in any club. The inventors of these techniques didn’t have professional societies looking over their shoulders and checking their work for conformity to standards. The same problems should be addressed in the same spirit today.

### Chunky data

“Chunky data” is a real problem. I remember a case where operators had manually measured the thickness of a plate at its four corners, with calipers that gave four significant digits, which they had rounded to two. Unfortunately, the information we were looking for was in the digits they had rounded off. We didn’t, however, need an mR chart to find out. It was visible in the paper spreadsheet. The risk of this happening is almost entirely eliminated in automatic data acquisition from sensors, instruments, and IIoT devices.

#XmR, #SPC, #ProcessBehaviorChart

Lonnie Wilson

March 2, 2019@ 9:21 amMichel,

Sweet derivation, not sure I could do those anymore. However, long ago I learned those fudge factors were based on the normal distribution and I am sure I could find it in some book of mine. However the person who said “…that the X-mR chart is a wonderful and currently useful process behavior chart, universally applicable, a data analysis panacea, and requiring no assumption on the structure of the monitored variables….” is not living in any world I have seen that might create data usable on a control chart.

A few years ago, like maybe 20…Donald Wheeler started hard selling the XmR and as Wheeler is a really bright guy, he could use it properly for for variety of data. That however is not true for eveyone. The interpretation of any control chart IS NOT INDEPENDENT of the raw data used. However in my practical experience the underlying assumption of normality is infrequently a limitation to process improvement.

This is due to two factors. Both the quality of the data and the quality of the process analyzed are so far from “good” that you do not need a very sharp instrument to make progress. However if you continue to work with that data and work with that process, it is important to understand the limitations of the tools you are using. You can carve a turkey with a 10″ kitchen knife or a surgeon’s scalpel ….. likewise you can take out an appendix with a either tool.

If you are going to use these tools, it usually works best if you understand them. I find that thought to be not only revolutionary but counter cultural to many trying to solve problems today…..

Nice posting

My 2 cents worth

Judy Dayhoff

March 2, 2019@ 9:58 pmHow about averaging 5 or more data points and using the central limit theorem to make the average from a normal distribution?

Michel Baudin

March 5, 2019@ 5:05 pmThis is just about working out the math behind a tool other people recommend. It’s about understanding it, not promoting or discouraging its use.

Lonnie Wilson

March 15, 2019@ 8:37 amI read this comment and I said, “Who is this guy and where has Michel Baudin gone??” I have known Michel for about 10 years mostly through blog postings and later when he proposed to exchange books we had respectively written. To me he was a very practical, hands on type of guy, so all this discussion of the math theory as it applied to a process improvement tool — totally threw me off. I could not believe what I was reading and I went on a journey to find my friend Michel. Lo and behold I found him in his bio, which obviously I had not read carefully enough, — it says, “trained in …. applied math”.

So now that I have found my friend I will bow out of this discussion, because although all this theory is interesting, when it comes to process improvements, it is largely, not completely, but largely irrelevant.

Michel Baudin

March 16, 2019@ 7:16 amYou caught me. I have been a closeted mathematician for more than 35 years. After I pivoted to manufacturing in the early 1980s, I noticed quickly that being open about this background was not helping me be effective. Judy Dayhoff was a colleague of mine at the time. Yet I kept using it without saying so, believing that nothing is as practical as a good theory.

Many years later, one comment my editor at Productivity Press made about Working With Machines was that it started with building block and combined them logically into systems. “It’s like math,” he said. Pascal Dennis also noticed that, in Lean Logistics, the discussion of supplier-customer relationships is based on the theory of the Prisoner’s Dilemma from game theory. I didn’t say so in the book to avoid spooking my readers.

Some years later, on a project, I noticed an error in a formula used in a client spreadsheet to calculate safety stocks. It was a basic typo — one term that should have been squared was linear and vice versa — but it made the formula produce consequential nonsense. I was not familiar with that formula, and couldn’t find a derivation of it anywhere. So I worked it out and posted it in this blog. To my surprise, it’s been the second most popular post for years.

Times have changed, and so has technology. I have come out of the closet. I hope it doesn’t cost me your friendship.

Renaud Anjoran

March 3, 2019@ 12:20 pmOh, yes, data are assumed to be normally distributed. Not the *underlying* data, but the means of those data (that’s why it’s called “Xbar” or “Xm”. See https://en.wikipedia.org/wiki/Central_limit_theorem.

I am not sure it’s true in the vast majority of cases, but I would tend to believe it’s true in most cases.

Or did I miss something? I am certainly no authority on this topic.

Matt Kelly

March 5, 2019@ 2:59 pmGreat read Michel. I am currently doing similar calculations in my Customer Analytics course. I also want to use these equations as they may help me with a school project I am working on involving an inefficient cleaning process. I will be back for more, Thanks !

Scott Hindle

March 14, 2019@ 4:40 amIn this free-to-access paper , Dr. Wheeler delves into the robustness of the d2 values that leads to the scaling factor of 2.66 (where 3/1.128=2.66) stated towards the start of this paper.

Hence, while the normal distribution provides the theoretical value of 1.128 for d2, Wheeler discusses what departures from normality mean in practice, not theory, where we have a limited no. of data and not an infinite amount.

I find XmR charts tremendously useful. The trick is in knowing how to use them to good effect. The “way of thinking” is the hard part I find. For example, how frequent should data be collected to achieve a meaningful, insightful XmR chart? The importance of this question is, I think, too little appreciated. Get the answer wrong and the XmR chart can be more problematic than helpful.

Michel Baudin

March 14, 2019@ 4:17 pm[latexpage]

If you don’t mind, I prefer to call the distribution “gaussian” rather than “normal” because of the implication in everyday speech that every other distribution is, in some way “abnormal.”

What I show in my post is that you can derive the control limits of the XmR chart straight from the properties of the gaussian distribution without introducing the or .

In his paper, Wheeler considers a limited range of distributions. They are all unimodal and just have nonzero skewness and kurtosis. The universe of possible product characteristics, however, is not this restrictive. Following are a few examples you might encounter:

Scott Hindle

April 2, 2019@ 3:04 amHi Michel, I would hope nobody thinks an XmR chart is for “everything”. Like all “tools”, use it to bring value / clarity / insight… if not, the use may be unnecessary.

No doubt different people will prefer certain tools over others. I have just found, as said, that an XmR chart is tremendously helpful.

If we have output from two machines with two modes, but both have the same nominal value (target), see how to correct this and bring them back in line with each other. A time series or histogram may be sufficient to identify the problem and help define a good way forward / action plan.

I’d argue that key to XmR charts is “rational sampling”, not worrying about normality.

Thanks!

Lonnie Wilson

March 16, 2019@ 8:35 amAu contraire Michel, it is refreshing to find a renaissance man amongst my many lean geek friends. For Dr. Deming, there was no “profound knowledge” without an understanding of the underlying theory. What continues to amaze me is that practitioners, often very soft on the needed skills, but who are poised, polished speakers and have a fine vocabulary of lean terms can often sway the many neophytes they deal with. However, when the rubber meets the road, and they must actually perform, they find they come up short. They then need to “get back to the books” OR find a convenient excuse for the non performance. Which reminds me of one of my favorite quotes, by John Kenneth Galbraith, who said,

“In the choice between changing one’s mind and proving there’s no

reason to do so, most people get busy on the proof”.

And for those folks who are weak on the theory, I find they become exceedingly creative when it comes to “getting busy on the proof”.

I agree fully with Dr. Deming that you must understand the underlying theory of your trade, be it supply chain management, lean applications or skill sets like engineering, sales, supervision and management. If you do not understand the underlying theory you will be limited to the writings documenting what others have accomplished – under the limited environment in which they accomplished that. Almost surely your circumstances will differ by some degree. Well, you need to understand the underlying theory, that is, if and only if, you want to be good at your trade. Success is not mandatory.

Losing a friend, hardly, I just have a renewed appreciation for an old one. Be well

Michel, sorry this has little to do with the derivation of d2, but I could not resist… and again, be well

Dr Tony Burns

March 28, 2019@ 6:42 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:43 amThe vocabulary of the field has changed so much since those days that I have no idea what Shewhart meant by this. I am sure he meant something specific with “statistical universe” but I have no idea what. On the other hand, the math shows exactly how limits are calculated, whatever interpretation anyone wants to put on it.

Dr Tony Burns

March 28, 2019@ 6:44 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:47 amThe XmR charts plots individual values, not averages. The Xbar chart plots sample averages. The primary reason is to dampen fluctuations so that shifts in the mean stand out better.

The Central Limit Theorem (CLT), as you said, does make sample averages converge to a Gaussian, provided the individual values are independent and have a mean and a standard deviation.

The Statistical Quality establishment, however, denies that it has any relevance. I don’t know why.

A serious problem with relying on sample statistics is that you need samples. With one-piece flow, you don’t want to wait until you have 5 pieces. You use go/no-go gauges on every part to react immediately and mistake-proof your processes.

It only works if your quality is high enough. If your first-pass yield is <97%, it won’t work, and neither will one-piece flow.

In this case, you should focus on yield enhancement, which requires more advanced techniques than control charts.

Renaud Anjoran

March 28, 2019@ 6:48 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:45 amIn Shewhart’s Statistical Method From The Point Of View Of Quality Control (1939), I don’t see anything but an effort to apply the state of the art in probability and statistics as it was in his day.

Admirable though his work is, parsing his words from today’s vantage point is a scholastic exercise of limited value. Our time is better spent researching and advancing the current state of the art.

The bottom line is that SPC control charts for measurements have control limits based on the assumption that, while under statistical control, the measurements are the sum of a constant and fluctuations that form a Gaussian white noise.

However you wish to interpret what it means, that’s the math, and anyone can check it without quoting any authority.

Dr Tony Burns

March 28, 2019@ 6:42 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:43 amThe vocabulary of the field has changed so much since those days that I have no idea what Shewhart meant by this. I am sure he meant something specific with “statistical universe” but I have no idea what. On the other hand, the math shows exactly how limits are calculated, whatever interpretation anyone wants to put on it.

Scott Hindle

April 2, 2019@ 3:24 amShewhart meant that SPC starts doubtful that a process will be “in control” (i.e. stable or predictable process). Without evidence of a reasonable degree of control (i.e. stable process) there is simply no probability model to talk about, neither gaussian nor other.

As an example, hypothesis tests, on the other hand, assume a stable process (e.g. iid data) and proceed without bothering to check this assumption/requirement… Yet, an unstable process undermines the interpretation of the hypothesis test.

While times have changed, and we should change with the times, the above holds true long after Shewhart’s pioneering work. When something new is better, embrace it. But if not, why change? Here we have a judgement call, people will see things in different ways…

Dr Tony Burns

March 28, 2019@ 6:44 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:45 amIn Shewhart’s Statistical Method From The Point Of View Of Quality Control (1939), I don’t see anything but an effort to apply the state of the art in probability and statistics as it was in his day.

Admirable though his work is, parsing his words from today’s vantage point is a scholastic exercise of limited value. Our time is better spent researching and advancing the current state of the art.

The bottom line is that SPC control charts for measurements have control limits based on the assumption that, while under statistical control, the measurements are the sum of a constant and fluctuations that form a Gaussian white noise.

However you wish to interpret what it means, that’s the math, and anyone can check it without quoting any authority.

Renaud Anjoran

March 28, 2019@ 6:46 amComment on LinkedIn:

Renaud Anjoran

March 28, 2019@ 6:46 amComment on LinkedIn:

Michel Baudin

March 28, 2019@ 6:47 amThe XmR charts plots individual values, not averages. The Xbar chart plots sample averages. The primary reason is to dampen fluctuations so that shifts in the mean stand out better.

The Central Limit Theorem (CLT), as you said, does make sample averages converge to a Gaussian, provided the individual values are independent and have a mean and a standard deviation.

The Statistical Quality establishment, however, denies that it has any relevance. I don’t know why.

A serious problem with relying on sample statistics is that you need samples. With one-piece flow, you don’t want to wait until you have 5 pieces. You use go/no-go gauges on every part to react immediately and mistake-proof your processes.

It only works if your quality is high enough. If your first-pass yield is <97%, it won’t work, and neither will one-piece flow.

In this case, you should focus on yield enhancement, which requires more advanced techniques than control charts.

Renaud Anjoran

March 28, 2019@ 6:48 amComment on LinkedIn:

Dr Burns

April 2, 2019@ 3:35 amYou claim “the measurements are the sum of a constant and fluctuations that form a Gaussian white noise. ” This is incorrect. Dr Wheeler proved Dr Shewhart’s assertion by showing that control charts work for 1143 different non Gaussian distributions, in his book “Normality and the Process Behavior Chart”.

Michel Baudin

May 29, 2019@ 10:35 pmIt is what the math is based on. It doesn’t mean you can’t use it in a broader context. No real gas is ideal, yet chemists routinely apply the ideal gas law.

The point of working out the math behind a formula is not to restrict its use but to understand how far you are straying from its underlying assumptions.

Antonio R Rodriguez

May 6, 2020@ 3:46 amRead Economic Control of Quality of Manufactured Product (Shewhart, 1931) and Statistical Method from the Viewpoint of Quality Control (Shewhart, reprint 1939). Dr. Donald J. Wheeler has published many books on the subject that clearly explains and address many misconceptions (e.g., normality, central limit theorem) for example (many more) Understanding Statistical Process Control, Advanced Topics in Statistical Process Control, Guide to Data Analysis.

Michel Baudin

May 6, 2020@ 6:11 amWhat exactly makes you think I am unaware of the writings of Shewhart or Wheeler?

Garrett

April 19, 2021@ 8:01 pmHi Michel,

I came across your article while seeking to understand control charts better as I am not one happy with plucked constants from thin air. I seemed to have missed a connection here in your derivation when you are calculating the mean. Could you possibly break that out a bit further on where you pull the mean of the range for the half gaussian?

Thanks,

Garrett

Michel Baudin

April 19, 2021@ 8:39 pmIf you want more details on the Half-Gaussian, please check out the Wikipedia article about, to which I linked in the post.

Garrett

April 20, 2021@ 6:34 pmAh this was a reading comprehension error. I read the linked wikipedia but for some reason did not catch that. In any case thanks for this article much appreciated and grants some nice understanding.