Last year I discussed the problems of transforming data prior to analysis (see my August 2009 column, “Do You Have Leptokurtophobia?,” my September 2009 column, “Transforming the Data Can Be Fatal to Your Analysis,” and my October 2009 column,“Avoiding Statistical Jabberwocky.”) There I demonstrated how nonlinear transformations distort the data, hide the signals, and change the results of the analysis. However, some are still not convinced. While I cannot say why they prefer to transform the data in spite of the disastrous results described in the three columns, I can address the fallacious reasoning contained in the current argument used to justify transforming the data. Since the question of transforming the data is closely associated with the idea that the data have to be normally distributed we shall discuss these two questions together.

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## Comments

## Strawman Argument

Dr. Wheeler is, as he frequently does, attacking a stawman on this topic.

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No one says data must be normal, or be transformed, before using a control chart. That would be silly, and for a very simple reason.

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The purpose of a control chart is to identify a process that is out of control, and if it is, it is *by definition* not a single process and therefore by definition cannot be normal (even if it could "pass" a test for normality). As we all know, the purpose of a control chart is to identify an out-of-control process so that you can eliminate these special causes of variation, not include them in a statistical model in order to get a chart that looks like a process that is in control! Now, if your data are too chaotic you might filter out wild outliers or you might use limits calculated from the median of the dispersion metric in order to establish better control limits for the underlying process to make identifying those special causes easier, but that is another discussion.

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So I'll arrogate upon myself to state that we all agree: You do not need to have, and in fact do not expect to always get, normal data when using control charts. You can use, and expect to use, control charts in the presence of non-normality.

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(The is one exception where you might correct control limits for non-normality: individual charts. Due to their sensitivity to distribution shape, assuming normality when the distribution really isn't inflates the probability of making an incorrect decision by a significant, but unknown amount of systematic error. Reasonable people will disagree on what to do, but when justified, I'll adjust control limits for non-normality on individuals charts using a very specific procedure to protect against systematic error inflation. Other people can live with an unknown and not quantified amount of risk or think the adjustments add more systematic error than they remove.)

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Of course, as implied by Dr. Wheeler above, you must test for normality before performing certain statistical tests (for the reasons I'll outline in this month's Six Sigma Heretic article) to make inferences, or when calculating process capability. Even then one doesn't "need" normality - there are many long-established ways to handle non-normality depending on the situation (the least attractive of which is a mathematical transformation). Just as one wouldn't use a t-test on ordinal data, there is a reason the rules are there and should be followed. I talk more about that in my column this month.

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It would be unjustifiably self-centered of me to assume that Dr. Wheeler is posting this in response to my article last month ( http://www.qualitydigest.com/inside/six-sigma-column/making-decisions-n… ) and since that article was about making inferences using hypothesis testing and the lovely benefits of the Central Limit Theorem, this article really wouldn't be a response to my article anyway.

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So I am assuming it is not.

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For this article to be in response to it, one would have read the title of my article without actually reading the article, and my experience with Dr. Wheeler is that he would not do that. But I thought to clarify for others who might not. Dr. Wheeler and I disagree on a few things, agree on far more, and when we disagree, he has always been courteous enough to spend the time to read the article.

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I am still waiting to meet those evil statisticians that say you must have normal data before using a control chart, though. Haven't run into one yet. So I do wonder to whom Dr. Wheeler is warning us against in these articles. Has anyone else run into one?

## Non-Normality in Data

Surely Dr. Wheeler jests if he expects anyone to claim that factory measured data will be neatly homogeneous and without outliers and worse, to expect a Shewart Chart with 3 sigma control limits to be sufficient and accurate enough to analyze data most of the time (97% to 98% +). Reality is, in phase I of SPC analysis, you will get a outliers and out of control points. The Cp, Cpk determines if the process is in control. If it is not, you fitting any kind of distribution to the data is less meaningful because it is unstable and should be viewed only to see unstability; we do not take the control limits seriously until the process is IN CONTROL. YOU CAN still fit a distribution JUST TO SEE HOW UNSTABLE IT IS IF YOU HAVE A PROGRAM TO DO IT. A statistical program WILL reveal outliers and Cp, Cpk values telling you the data is unstable. The chart is meaningless except for the Cp, Cpk values and outliers detected. Go back and fix the common causes that are producing the outliers. We are not trying to fit or establish a model here, we are merely checking for stability and control. Non-homogeneous data is only one cause of instability, not the sole cause of it. Fix the product and go back to see if the data is now stable and in control. If it is stable, NOW is the time to fit a distribution to the data. Assuming normality on all homogeneous data and "sufficing" to fit a NORMAL distribution to everything is a mistake. I disagree with Dr. Wheeler completely on this. Once outliers problems are fixed, the data is ready to be fit by THE BEST DISTRIBUTION THAT FITS IT and NOT by assuming normality on all data, even if it is homogeneous.

If you have a statistical program that fits a statistical distribution to the data, use it, it will not hurt nor is it a waste of time. The KS or AD tests are statistic examples that can be used to test for the best distribution using a program. Then use the best-fit distribution to compute the two-sided or one-sided control limits. Fitting data with the best probability distribution will reduce false alarms over fitting normal distributions all the time. True, some data will have control limits nearly the same if fit by more than one distribution. But in my case, having analyzed over 2000 sets of variable data, I have found that a non-normal distribution best fits the data 70% of the time. If you have an automated SPC program to compute SPC charts, Cp, Cpk, Yield (Performance), test for outliers, etc., you will be far ahead of the game. There are many examples that show if for example, if you fit a normal distribution to lognormal data, your control limits, Cp, Cpk, and Yield will be significantly different. If any distribution fit, even a normal one, is applied to the data, outliers will skew the data significantly; the SPC may present a picture that will have some value (won't hurt), but the Cp, Cpk and outliers are the important factors to consider to explore a common cause.

We DO NOT EXPECT DATA TO BE NORMAL or homogeneous; we test data for stability and check for outliers, fix the problem(s) and regenerate the data for SPC analysis again. After no ouliers are seen and the process is stable, we fit the BEST DISTRIBUTION to the data in case the data is significantly skewed and re-evaluate it for control. Once control has been established, we set those limits for Phase II SPC analysis. The Yield is simply the area under a best-fit distribution between the spec limits (assumiing two-sided spec limits) and for acceptance testing it should be nearly 100%. Anything less is considered a probablity of nonconformance (PNC = 100% - Yield). If PNC is significant, the product component needs rework before it goes to the customer.

## Evil Statiticians

I hace an article by Lloyd Nelson called "Notes on the Shewhart Control Chart". In the article he lists five statements/myths about the charts including:

c) Shewhart charts are based on probabilistic models

d) Normality is required for the correct allpication of an Xbar chart.

He then goes on and states"Contrary to what is found in many articles and books, ......these statements are incorrect"

If someone as well known and respected as Dr. Nelson makes a statement like that, I take it to me true.

I wouldn't say these statisticians are evil, just mis-informed. SPC was based on economic considerations not just probability/mathamatical theory.

Nelson concludes his article by stating ".....wrongly assuming that Shewhart derived the control chart by mathematical reasoning based on statistical distribution theory. His writings clearly indicate that he did not do this"

Rich DeRoeck

## Interesting Discussion

This article is in large part an excellent summary of some of the concepts in Don's book "Normality and the Process Behavior Chart."

David, Don is not suggesting that “factory measured data will be neatly homogeneous and without outliers;” in fact, he stated that his experience is that 90% of processes will be out of control when you look at them. I’m not sure what the rest of your argument is, though. Are you really saying that the “Cp, Cpk determines if the process is in control?” Are you suggesting that the Cp or Cpk have any meaning if the control chart shows that the process is unstable? In one section of your response, it seems that you are claiming that.

Of course, you can’t rely on the control limits when there are out-of-control points or non-random patterns. You have to work to get the process stable before you can use those limits for prediction. You have to have the control limits, though, to know whether the process is stable or not. Trying to fit a distribution to a set of data that have been shown (in a control chart) to be out of control is like trying to calculate the standard deviation of a set of phone numbers…you can do it, but it doesn’t prove anything because it doesn’t mean anything. There is no meaningful distribution for a set of data that are out of control.

Frankly, I don’t know what to do with statements such as “The chart is meaningless except for the Cp, Cpk values and outliers detected. Go back and fix the common causes that are producing the outliers.” Which chart? Are you talking about a capability six-pack from Minitab? Most of the process improvement world is in agreement on one thing: common causes don’t signal as outliers. Usually we try to fix the special or assignable causes.

I agree with much of what’s in your last couple of paragraphs; I’m still not sure that there’s a need to fit a distribution. It is probably worthwhile to test for non-fit, if you’re going to use some test that requires some particular distributional assumption. It would also be useful if you plan to do some Monte Carlo simulations, and want to get parameters for a best-fit distribution in your simulations.

Steve, I don't think Don is responding to you in this article. He's more likely to be responding to Forrest Breyfogle and numerous other Six Sigma practitioners who love to test for normality and transform any non-normal data before they include them in any analysis (including control charts). While most of what Forrest has written has been aimed at tranforming in the case of individuals charts, there can be little doubt that there is an over-emphasis on normality (even in individuals charts), and transforming data generally. It's not at all difficult to find articles or presentations that will tell you to transform your data before placing it on a control chart. I have met people who have told me (and published in their courseware) that the transformation of data via the central limit theorem is "the reason that Xbar-R charts work."

This is hardly a new argument. In the Control Charts menu of Minitab, the first entry is for the Box-Cox Transformation. Why? In the help function for that transformation, it states “Performs a Box-Cox procedure for process data used in control charts.” It also then goes on to quote Wheeler and Chambers on why transformation is probably not needed, but the fact that it’s there, and stated that way, indicates that it’s not an uncommon argument.

Some people just love to transform. I recently saw (in a book purporting to be for preparation for the ASQ Six Sigma Black Belt certification exam) a logarithmic tranformation, to transform a binomial proportion into a Poisson rate, so you could estimate a DPMO! So I transform a proportion non-conforming to an estimate of a Poisson rate, so I can use that to estimate a proportion non-conforming? Another clear victory for calculation over common sense!

It's not just confined to Six Sigma, though. I heard the same argument from statisticians years ago, well before Six Sigma became popular. Many statistics texts and courses don’t cover analytic studies at all; some may cover control charts at a superficial level as an afterthought; this includes texts for business and engineering stats at the undergrad and graduate levels. Most of the literature around ImR (XmR) charts and non-normality relies on the ARL argument; the ARL portion of this article is, I think, a counter to that argument. I heard Don make this argument in a presentation at the Fall Technical conference in 2000; it was well-received by a large audience of statisticians.

## Following comment is sincere!

Hi Rip,

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No, I really don't think that Dr. Wheeler meant this as a response to my article. It is difficult to convey sincerity in text, I guess. But the first comment I had to my article DID assume that what I was saying was that you needed normality for using control charts, so I wanted to be clear that people did not misunderstand.

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The Central Limit Theorem is quite different than a transform of the raw data, and it is certainly comes to bear in SPC (the reason we don't put spec limits on an X-bar chart, for example, or how taking a bigger sample affects our ability to detect more subtle shifts, or calculating sample size all relate to the CLT). The concept of the Random Sampling Distribution (in this case of the ranges or standard deviations) is used to get the estimate of the variance of the underlying process from the within-sample dispersion, which is then used to generate the control limits for the location chart. That is the real genius of control charts - the within-variability should be the best you can hope for from the process (since the samples are sequential) and so should more closely represent the underlying process variability than, say, actually looking at the dispersion of all the data points.

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I get such a kick out of walking my students through that...

## Thanks for the comments

My thanks to Rich DeRoch and Rip Stauffer for their insightful comments. I had forgotten Lloyd Nelson's comments on this topic. In answer to Steven Ouellette's question, there are many statisticians, and many engineers who take as an article of faith that the data have to be normally distributed or else they have to be transformed to make them approximately normally distributed prior to placing them on a process behavior chart. If you have not run into these, then count yourself fortunate.

Unlike politics, simply repeating an untruth over and over again does not make it true. I have explained my position very carefully here. If my argument is wrong, then point out where I have erred. If my argument is correct, then the result is correct, regardless of how unpalatable it may be for some. Rants are simply not acceptable as mathematical arguments.

Donald J. Wheeler, Ph.D.

Fellow American Statistical Association

Fellow American Society for Quality

## My 2 Cents' Worth

In 1991, I attended Dr. Deming's 4 day seminar in Cincinatti. I distinctly heard him make the following statement: "Normal distribution? I never saw one." At the time, I was astounded and that's why I remember it so well. However, now I understand (at least partially) what he meant.

Dr. Wheeler has always provided elegant proofs (theoretical and practical) of his statements such as found in this article. In addition, his statements are founded in the work of Dr. Shewhart, who, of course, invented (or discovered) the process behavior chart. What more does anyone need to understand that the data do not have to be "normally distributed" for a process behavior chart to give useful insight into the process.

After over 25 years of learning and practising the use of process behavior charts in industrial settings, I can honestly say that I have never had to worry about how the data was distributed when using I-mR charts; and I have helped improve many processes this way. I have also learned that the simplest analysis which gives you the insight needed to solve problems is the best analysis. Start transforming data and trying to lead a team to a solution to a problem and all you will get is glazed-over eyeballs!!! Keep it simple, stupid and people will follow you down the road to continuous improvement. Start transforming data and using other confusing "statistics" and people will stay home!

Back to the normal distribution: Keep collecting data from a stable system after determining that the data follows the normal distribution....and eventually, the data will fail the so-called goodness of fit test. In addition, almost any set of data that fits one distribution model will fit others as well. Normal distribution? I never saw one either (though I used to think I had)!!!

## Lloyd Nelson's Article

Does anyone know where I might obtain a copy of Lloyd Nelson's article?

Steve Moore

smoore@wausaupaper.com

## Non-Normality in Individual Type Data

According to Dr. Borror, Dr. Douglas Montgomery, and Dr. George Runger, individuals type SPC charts are sensitive to non-normality and you will have false alarms if you assume normality. Tom Pyzdek's article gave an excellent example of that in comparing control limits computed using a normal and a lognormal distribution; the control limit differences are very significant.

For Phase II performance of the Shewart chart, Dr Montgomery states in his "Introduction to Statistical Quality Control" that in-control individual type control charts are sensitive to non-normal distributions. I am not stating anything about X-MR charts with multiple subgroups, only individual types. The suggestion is to do one of three things: (1) use percentiles from the best-fit distribution to determine control limits (2) Fit a distribution that best fits the data using the KS or AD statistic to compute control limits, or (3) transform the data to determine proper control limits. I think the Borror, Montgomery, and Runger studies show evidence that we must fit the best distribution to individual in-control data to compute control limits. I noticed Dr. Wheeler never mentioned anything about individual SPC charts and their sensitivity to non-normality. Why do you think there are statistical programs including these features in statistical software to fit the best distribution to the data, such as MINITAB and many others? If the data is not in control in Phase I, I think we all agree that SPECIAL or assignable causes should be investigated. Using statistical software that fits individual data that contains outliers does not hurt if you generate SPC charts and certainly is not a waste of time. Control limits may not mean much but you can always see suspect outliers skewing the data. Fitting data that contains outliers is not to be taken seriously; I previously said go back and eliminate special causes and redo your SPC chart. Out of control data can be easily seen if you fit the best distribution or assume normality. The object is to explore the data for stability and control before and during phase I. And by the was Rip, Cp, Cpk do not determine stability; Cp and Cpk only have to do with how well data is meeting spec and is meaningless if the process is NOT under control. Both the YIELD and Cp, Cpk are meaningful only after the data is under control and stable, ready for phase II. If I mis-stated myself earlier, my apologies, I stand corrected.

Box-Cox transformations such as ones that MINITAB uses ARE appropriate and do not distort the data for individuals charts. In computing Yield, Dr. Jerry Alderman and Dr. Allan Mense created a program that fits a Johnson distribution to data to determine the Yield. The results were run on thousands of datasets and the results are excellent. Not all data can be fit by a Johnson distribution, but each indicates reasons why. Results were compared to Yields computed using MINITAB and JMP 7. Results are very good. However I prefer to fit the data by the proper distribution then compute the Yield without transforming the data in any way. To the person that said it is meaningless to fit data to telephone book numbers, you need to go back to school; we deal with fitting measured data, not a bunch of meaningless random numbers. Measured data are not random.

Why do I pick on individuals SPC charts? Because this is the only way data is measured in our factories. We may change that process for justifiable reasons in the future, but for now, all data comes as individuals over time. One comment here was made to assume normality for all data and keep it simple stupid. That is a narrow minded approach inviting risks of false alarms and unnecessary rework, including shutting down a line to investigate a false alarm. For Phase II SPC processes just use software that best-fits your data, computes proper control limits, yields, Cp, Cpk, and SPC charts. How do you handle data that is discrete, e.g., data that has a few or many step functions or constant values with a few outliers? Fitting any distribution including a normal distribution and assuming normality is improper. I welcome Dr. Wheeler to respond to the Borror, Montgomery, Runger studies on non-normality of individuals SPC charts. Dr. Montgomery and Dr. Runger each suggest that a univariate EWMA chart would be insensitive to non-normality and could be used if the right lambda and L values would be used. Those non-statisticians doing SPC have no business generating or evaluating SPC charts. Software makes computations and the statistics invisible to the SPC analyst; but the person analyzing them had better know basic statistics of SPC charts. This is the 21st Century, not the 20th Century. Also no one in their right mind computes SPC charts and Yields by hand. Yield = area under the curve between spec limits once a process is under control. Percent nonconformance = 100% - Yield.

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