Statistical Process Control

There are not so many articles who really explain Deming’s methodology well because, though he is the founder of Quality with Walter A. Shewart, most of the “modern Quality Gurus” do not really understand his strange mixture between Management, Philosophy and Statistics. That’s why I’m so pleased to be able to read this article:

Becoming a Black Belt without Six Sigma

by William H. Goodenow

Quality Assurance Manager

One of W. Edwards Deming’s major contributions to the accumulated body of knowledge dealing with experimental design is his differentiation between enumerative and analytic studies. The key difference between the two is that the first deals with static (snapshot) conditions, while the latter deals with dynamic (changing) conditions. It is because of this fundamental distinction that the use of design and analysis tools which are intended for static conditions (traditional statistical methods) can be both inappropriate and ill advised for dynamic conditions.

There are other reasons for this concern. For example, most traditional methods of statistical analysis require that certain assumptions be met, the most common are:

• Normality of the data (a unimodal & symmetric normal–bell-shaped–distribution)
• Equivalence of the variances (equal variability in the test or response data)
• Constancy of the cause system (all non-controlled factors held constant)

These assumptions may or may not be met, or even tested for validity, before the analyses are performed–especially with the simplicity of using many of today’s off-the-shelf statistical software packages.

In addition, Deming recognized that most symmetric (or composite) functions of a set of numbers almost always throw away a large portion of the actual information contained in the data. A traditional statistical test of significance is a symmetric function of the data. In contrast to this, a proper plot of data points will conserve the inherent information derived from both the comparison itself, and the manner in which the data were collected. For example, symmetric functions, such as the mean and standard deviation, can easily gloss over unknown and unplanned changes in the cause system over time, and, consequently, make a big and potentially misleading difference in the message the data are trying to convey for the purpose of providing a reliable prediction of future process behavior.

In other words, both the “design” approach and the methods of analysis typically taught and used for industrial experimentation leave much to be desired from a reliability of prediction point-of-view.

Design

Since the purpose of industrial experimentation is to improve a process and product’s performance in the future, when some conditions may have changed, design efforts must provide for conducting the study over an appropriately wide range of conditions.

The actual determination of how wide to make this range is a critical part of the design stage of experimentation. If the range of conditions selected is too wide, the DOE team could falsely conclude that observed changes in the process will continue in the future when, in fact, these conditions may not be operationally feasible in the long run. If the range of conditions is too narrow, the team may miss important improvements that could result under a wider range of conditions. These errors are not quantifiable in analytic studies, and the reliability of any conclusions reached regarding future performance will be a function of how closely we follow Deming’s design principles.

Other considerations include choosing the best variables for a study; handling background variables (those conditions to be either held constant or varied in an appropriately controlled manner), and nuisance variables; unknowns which can be neither held constant nor varied in a controlled manner; and deciding on replication, methods of randomization, the Design Matrix itself, planned methods of statistical analysis, cost and schedule.

There are still two considerations to be addressed:

What is the objective of the study? What background information do you already possess about the variables under study, in terms of their individual descriptive properties (mean, mode, median, standard deviation, the shape of their respective distributions, etc.), and the relationships between them.

Before establishing the objective, we must be as candid and complete as possible about the knowledge we already possess.

The problem with the “planners plan and doers do” mindset is the artificial separation it creates between the planning and execution steps of problem-solving. Deming recognized this tendency and much of his management and statistical thinking was directed at overcoming it. In Figure 1, we see how the idea of sequential learning not only applies to DOE, but how the process of stepwise experimentation, when properly used, can lead to the acquisition of profound knowledge.

To insure that the design process adequately provides for the execution of an appropriate and reliable experiment, it will be helpful to actually use some sort of design checklist and worksheet.

The reasons

A clue to the reasons why traditional analysis methods leave much to be desired in an analytic study was given in our discussion of the problems with symmetric functions alluded to by Deming and his followers–such as losing information inherent in the process and data collection activity. As troublesome as this tendency is, it does not represent the last straw. The fatal flaw shows up when future facts of life change sufficiently to render the original conclusions meaningless at best and clearly wrong at worst. It doesn’t matter how big the F-Ratio is in the ANOVA Table, or other tests of statistical significance, they have no meaning if the conditions under which they were derived no longer exist in the same proportions as in the original study.

The alternatives

The best way to analyze data from an analytic study is to use the old stand-by methods of charting. This may include control charts of the SPC variety, variations of these or even the integration of SPC and DOE.

Figure 2 shows a slightly different way of diagramming the sequential application of the Scientific Method (Figure 1). Alternating uses of SPC and DOE can provide a concrete way of developing and exploring various process optimization hypotheses (DOE), and confirming the efficacy of these operational models in the future with carefully planned process control charts (SPC). As the difference between our theories and the facts narrows, knowledge grows.

The really important aspects of control chart analysis are that it: takes into account the order in which the data were generated and collected; and does not require assumptions of normality, equivalence of the variances or constancy of cause systems, since the data in the experiment allows us to reject any or all of these hypotheses. It allows us to study the individual data points, groups of data points, and various patterns over time that may provide valuable clues as to why certain measurement points showed different results than others.

It is this last benefit that forms a fundamental and extremely powerful basis for the analysis of any analytic study. Not only would we have been misled by traditional statistics, but a cursory examination of the control charts, as well, could have allowed us to not really see what the data were trying to tell us.

About the author:

William H. Goodenow has been an examiner with the Wisconsin Forward Award (WFA) program, Wisconsin’s adaptation of the Malcom Baldrige Quality Award.

The existence of Normal Law is based on the Central Limit Theorem, that’s probably the reason why the huge majority of people - after they have followed some Statistics course at their Business or Engineering High School - believe that in the Real World, Normal Law is so common.

That’s a big myth: Normal Law especially in Manufacturing is very rare; in fact that’s the very foundation of Shewhart’s talk about a “controlled process“: if Normal Law was magic, there would be no need for him to invent this latter concept in his book “Statistical Method from the Viewpoint of Quality Control“.

It is astonishing that some practionners in Quality field do rediscover the wheel even after they have heard about Walter A. Shewhart - whereas others seem to have totally failed to even engage in the way to do so - I reproduce below an article By Thomas Pyzdek.

I’m not affiliated with him - I don’t know him whatsoever - and I only posted his opinions here to illustrate what I’m claiming above as well as to serve as a reference from my other blog’s article about “Shewhart/Deming Statistical Process Control vs Six Sigma“.

Non-Normal Distributions in the Real World
Copyright © 2000 by Thomas Pyzdek, all rights reserved
Reproduction allowed if no changes are made to content

One day, early in my quality career, I was approached by my friend Wayne, the manager of our galvanizing plant.

‘Tom,” he began, “I’ve really been pushing quality in my area lately, and everyone’s involved. We’re currently working on a problem with plating thick­ness. Your reports always show a 3-percent to 7-percent reject rate, and we want to drive that number down to zero.”

I, of course, was pleased. The galvanizing area had been the company’s perennial problem child. “How can I help?” I asked.

“We’ve been trying to discover the cause of the low thicknesses, but we’re stumped. I want to show copies of the quality reports to the team so they can see what was happening with the process when the low thicknesses were produced.”

“No problem:’ I said, “I’ll have them for you this afternoon.”

Wayne left, and I went to my galvanizing reports file. The inspection procedure called for seven light poles to be sampled and plotted each hour. Using the reports, I computed the daily average and standard deviation by hand (this was before the age of personal computers). Then, using a table of normal distribution areas. I found the estimated percent below the low specification limit. This number had been reported to Wayne and a number of others. As Wayne had said, the rate tended to be between 3 percent and 7 percent.

I searched through hundreds of galvanizing reports, but I didn’t find a single thickness below the minimum. My faith in the normal distribution wasn’t shaken, however. I concluded that the operators must be “adjusting” their results by not recording out-of-tolerance thicknesses. I set out for the storage yard, my thickness gage in hand, to prove my theory.

Hundreds of parts later, I admitted defeat. I simply couldn’t find any thickness readings below the minimum requirement. The hard-working galvanizing teams met this news with shock and dismay.

“How could you people do this to us?” Wayne asked.

This embarrassing experience led me to begin a personal exploration of just how common normal distributions really are. After nearly two decades of research involving thousands of real-world manufacturing and nonmanufacturing operations, I have an announcement to make: Normal distributions are not the norm.

You can easily prove this by collecting data from live processes and evaluating it with an open mind. In fact, the early quality pioneers (such as Walter A. Shewhart) were fully aware of the scarcity of normally distributed data. Today, the prevailing wisdom seems to say, “If it ain’t normal, something’s wrong.” That’s just not so.

For instance, most business processes don’t produce normal distributions. There are many reasons why this is so. One important reason is that the objective of most management and engineering activity is to control natural processes tightly, eliminating sources of variation whenever possible. This control often results in added value to the customer. Other distortions occur when we try to measure our results. Some examples of “de-normalizing” activities include human behavior patterns, physical laws and inspection.
Human Behavior Patterns

Figure 1 shows a histogram of real data from a billing process. A control chart of days-to-pay (i.e., the number of days customers take to pay their bills) for nonprepaid invoices showed statistical control. The histogram indicates that some customers like to prepay, thus eliminating the work associated with tracking accounts payable. Customers who don’t prepay tend to send payments that arrive just after the due date. There is a second, smaller spike after statements are sent, then a gradual drop-off. The high end is unbounded because a few of the customers will never pay their bills. This pattern suggests a number of possible process improvements. hut the process will probably never produce a normally distributed result. Human behavior is rarely random, and processes involving human behavior are rarely normal.

Figure 1. Days between mailing of invoice and receipt of payment.

Physical Laws

Nature doesn’t always follow the “Normal Law” either. Natural phenomena often produce distinctly non-normal patterns. The hot-dip galvanizing process discussed previously is an example. A metallurgist described the process to me (but too late, alas, to prevent the aforementioned debacle) as the creation of a zinc-iron alloy at the boundary. The alloy forms when the base material reaches the temperature of the molten zinc. Pure zinc will accumulate after the alloy layer has formed. However, if the part is removed before the threshold temperature is reached, no zinc will adhere to the base metal. Such parts are so obviously defective that they’re never made.

Thus, the distribution is bounded on the low side by the alloy-layer thickness, but (for all practical purposes) unbounded on the high side because pure zinc will accumulate on top of the alloy layer as long as the part remains submerged. Figure 2 shows the curve for the process - a non-normal curve.

Figure 2. The distribution of zinc-plating thicknesses.

Inspection

Sometimes inspection itself can create non-normal data. ANSI Y14.5, a standard for dimensioning and tolerancing used by aerospace and defense contractors, describes a concept called “true position.” The true position of a feature is found by converting an X and Y deviation from target to a radial deviation and multiplying by two. Even if X and Y are normally distributed (of course, they usually aren’t), the true position won’t be. True position is bounded at zero and the shape often depends solely on the standard deviation.

Many other inspection procedures create non-normal distributions from otherwise normal data. Perpendicularity might be normally distributed if the actual angle were measured and recorded. Quite often, though, perpendicularity is meas­ured as the deviation from 90 degrees, with 88 degrees and 92 degrees both being shown as 2 degrees from 90 degrees. The shape of the resulting distribution varies depending on the mean and standard deviation. Its shape can range from a normal curve to a decidedly non-normal curve. This apparent discrepancy also applies to flatness, camber and most other form callouts in ANSI Y14.5. The shape of the curve tells you nothing about your control of the process.
Implications

At this point, a purist might say, “So what?” After all, any model is merely an abstraction of reality and in error to some extent. Nevertheless, when the error is so large that it has drastic consequences, the model should be re-evaluated and perhaps discarded. Such is often the case with the normal model.

Process capability analysis (PCA) is a procedure used to predict the long-term performance of statistically controlled processes. Virtually all PCA techniques assume that the process distribution is normal. If it isn’t, PCA methods, such as Cpk, may show an incapable process as capable, or vice versa. Such methods may predict high reject rates even though no rejects ever appear (as with the galvanizing process discussed earlier) or vice versa.

If you’re among the enlightened few who have abandoned the use of “goal-post tolerances” and PCA, you’ll find that assuming normality hampers your efforts at continuous improvement. If the process distribution is skewed, the optimal setting (or target) will be somewhere other than the center of the engineering tolerance, but you’ll never find it if you assume nor­mality. Your quality-improvement plan must begin with a clear understanding of the process and its distribution.

Failure to understand non-normality leads to tampering, increased reject rates, sub-optimal process settings, failure to detect special causes, missed opportunities for improvement, and many other problems. The result is loss of face, loss of faith in SPC in general, and strained customer-supplier relations.

From 1703 until his death in 1705, Jacob Bernoulli exchanged a number of letters with Gottfried Leibniz. Leibniz wrote to him:

“La nature a sans doute ses habitudes, provenant du retour des causes, mais ce n’est que la plupart du temps. C’est pourquoi, ne peut-on pas objecter qu’une nouvelle expérience puisse s’écarter un tant soit peu de la loi de toutes les précédentes, du fait de la variabilité même des choses ? De nouvelles maladies se répandent souvent sur le genre humain et par conséquent quelque soit le nombre de morts dont vous avez fait l’expérience ce n’est pas pour autant que vous avez établi les limites des choses de la nature au point qu’elle ne puisse en varier dans le futur.”

“Nature has established patterns originating in the return of events but only for the most part. Therefore, can’t we argue that a new experience could deviate from the law of all the previous ones, even a little bit, because of the very essence of variability of things ? New illnesses often flood the human race, so that no matter how many experiments you have done, you have not thereby established a limit on the nature of events so that in the future they could not vary.”

Later, John Maynard Keynes directly refers to Leibniz in his essay on “Probability in relation to the theory of knowledge“. According to a Cambridge Journal’s article, “Keynes’s Treatise on Probability contains some quite unusual concepts, such as non-numerical probabilities and the ‘weights of the arguments’ that support probability judgements. Their controversial interpretation gave rise to a huge literature about ‘what Keynes really did mean’, also because Keynes’s later views in macroeconomics ultimately rest on his ideas on uncertainty and expectations formation”. But what Keynes really means was just what he once told clearly:

“By uncertain knowledge … I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty …

The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention … About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know!”

At the same time, during the 1920s, Walter A. Shewhart, statistician and engineer, was commissioned to improve the quality of telephones manufactured by Bell Laboratories.

Shewhart framed the problem in terms of Common and Special-Causes of variation. Though Shewhart may not have been the first to reveal this concept, he is the first who has established an operational mean to distinguish between the two: on May 16, 1924, he wrote an internal memo introducing Statistical Process Control with a Control Chart as a tool for Continuous Process Improvment.

• “Management is prediction.”
• “A goal without a method is nonsense.”
• “Without theory, there are no questions.”
• “The process is not just the sum of its parts.”
• “The problem is that most courses teach what is wrong.”
• “Monetary rewards are not a substitute for intrinsic motivation.”
• “Does experience help? No! Not if we are doing the wrong things.”
• “We should work on the process, not the outcome of the processes.”
• “Management by results is confusing special causes with common causes.”

Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification “width”) to the spread of the process values, as measured by 6 process standard deviation units (the process “width”).

The basics of Statistical Process Control (SPC) is to make the Process Control Limits fit within the Specifications Limits.

Control Limits define the predictable process variations. In practice it cannot be known without some associated probability.

Nevertheless, Control Limits in SPC, from the founder Shewhart’s point of view, should not be confused with Confidence Interval in Probability. Notably, it is a mistake to use Student t-distribution Confidence Interval to estimate SPC Control Limits.

The basics of Statistical Process Control (SPC) is to make the Process Control Limits fit within the Specifications Limits.

Variations due to special causes are

• localised in nature
• exceptions to the system
• considered abnormalities
• often specific to a
  • certain operator
  • certain machine
  • certain batch of material, etc.

Investigation and removal of variations due to special causes are key to process improvement

Variations due to common causes

• have small effect on the process
• are inherent to the process because of:
  • the nature of the system
  • the way the system is managed
  • the way the process is organised and operated
• can only be removed by
  • making modifications to the process
  • changing the process
• are the responsibility of higher management

Control charts are used to detect whether a process is statistically stable. Control charts differentiates between variations

• that is normally expected of the process due chance or common causes
• that change over time due to assignable or special causes