Archive for the ‘Statistical Tools for Six Sigma’ Category

The Future of Six Sigma 2013 Update

Tuesday, March 26th, 2013

I am often asked my opinion regarding the future of Six Sigma.

Regarding the future of Six Sigma, it continues on despite rumors of its death which began shortly after its birth in the 1980s. For those doing it right (arguably a minority, but a sizable one,) the Six Sigma approach has evolved into a new way to lead and manage an organization. Many have rebranded the approach to shed the baggage which Six Sigma has accumulated during its 27-year run as a “fad.” The new approach to leadership and management is distinguished from the traditional approach by four characteristics:

  1. a balanced approach to stakeholder demands (versus managing primarily for shareholders,)
  2. a balance of short- and long-term goals (versus a focus on quarterly results,)
  3. emphasis on facts and data, (versus reliance on expert opinion,) and
  4. a “horizontal” value stream perspective (versus a top-down command-and-control hierarchy.)

Any one of these things would be a game-changer. Taken as a whole they would ordinarily be thought of as revolutionary. However, probably due to the fact that the changes happened over nearly three decades, they haven’t been widely recognized as having the impact that they’ve had. Instead, as organizations using this approach have pushed their usage upstream to suppliers and downstream to customers, their adoption has slowly spread from United States manufacturers to all industries globally. As a result it is now commonplace for career guidance counselors to advise people to become Six Sigma certified. Some advise recipients of Bachelors degrees to become Six Sigma certified before pursuing Masters degrees.

The Next Big Thing: Big Data

One thing I’d like to see embraced by Six Sigma is the Big Data Revolution, which is a theory-free approach to using data in corporate data warehouses. Big Data is akin to part of the Measure Phase of a Six Sigma project, except that instead of using information in a data warehouse to test ad hoc theories, Big Data crunches the data warehouse contents to look for correlations. Correlations are then used for planning activities and, usually, the cause of the correlation is not pursued. This is very different than the use of data in a Six Sigma project, where the analysis is focused on achieving a particular goal. I don’t see Big Data as a competitor but as an opportunity for the Six Sigma community to move into another area. After all, analysis is a skill set Six Sigma practitioners have. We need to add a few new tools to our toolkit (e.g., data mining tools,) but these are similar to the statistical tools we already use .

Six Sigma and the quality profession can add a dimension to Big Data by filling in the gap between correlation and causation. By employing our ability to assemble interdisciplinary teams and utilizing the tools of experimental design, we can go beyond Big Data’s casual acceptance of correlation and answer the all-important question: why does this correlation exist? This is essential if we are to avoid the many traps that result from blindly acting on correlation without a deeper understanding of cause-and-effect. For example, a call center using Big Data discovered that callers who were kept on hold for as long as 1-hour were no less satisfied with their experience than callers whose calls were answered immediately, providing their issue was resolved. Further research into the cause of this unexpected result led to the determination that the missing variable was that many callers hung up rather than wait an hour for their calls to be answered. The customers who abandoned the call were not asked to complete the after-call survey. When these callers were contacted and their satisfaction scores added to the data, the  correlation not only disappeared, it was reversed. I.e., customer satisfaction declined as hold time increased.

Big Data also misses the boat in a number of other ways that Six Sigma and quality professionals can address. There are inherent problems with relying solely on data in data warehouses. These data are generally operational data, not data from planned experiments. Thus, they are often missing important variables. When variables are not manipulated in a planned way, statisticians are often not able to disentangle their interrelationships. They are also not able to properly explore important interactions between the variables. Operational processes are carefully controlled, so the variables involved don’t vary by much, leading to the “range restriction effect” that hides underlying relationships. These and other shortcomings of “happenstance data” analysis are well-known to Black Belts and Quality Engineers.

Speaking of skilled professionals, the obvious preferred group for addressing Big Data issues is Statisticians. However, Statisticians are in notoriously short supply and have been for decades (if not always.) Six Sigma “belts,” quality engineers, and reliability engineers are trained in a significant subset of useful statistical techniques. This pool of skilled workers can be leveraged to greatly expand the reach of the few statisticians available in most organizations.

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A Discussion of R-Square and S

Sunday, October 7th, 2012

Question from a Black Belt student

When selecting a model two of the criteria say that  the Standard Error be small and R Square be large.

Since R Square is a proportion, one might think that the usual 95% would be the threshold — correct?

My Response

It may be surprising, but most of the cutoff values used in statistics are quite arbitrary. This includes the p-value cutoff of 5% for hypothesis tests and confidence intervals. This discussion of p-values is important because in the discussions of R2 and S below I assume that the p-value for the model is “statistically significant.” In other words, I discuss the cutoffs for R2 and S values based on the assumption that both are from statistical models that meet an arbitrary cutoff for the model’s p-value!

R2

With R2 no arbitrary cutoff has ever become the accepted norm. Thus the advice that R2 should be “large.” In this case, large is in the eye of the experimenter.  Indeed, what is considered large varies a great deal according to the type of experiment being conducted. In social science experiments researchers are delighted with statistically significant R2 values as low as 0.2 or even lower. In hard science and engineering experiments R2 values greater than 0.9 is often expected. As a general rule, the more the researcher knows about the science, the better controlled the experiment can be and the expectation for R2 increases. Obviously, humans behavior is poorly understood, to the point where some question the usage of the term “Social Science.” So R2 values that physicists and engineers would dismiss out-of-hand are acceptable in that field.

In Lean Six Sigma we usually find ourselves somewhere in the middle of these two extremes. If our projects involve customer responses, then statistically significant R2 values around 0.5 might be enough to give us the direction we need for improvement. But if we are improving, say, cycle time through a a process, then our threshold would be higher, perhaps 0.7. Still, as you can tell, these are arbitrary. The point is that we want our data to point us in the right direction for making improvements. What R2 value will do this for us varies on a case-by-case and project-by-project basis.

S

The proper value for the standard error, S, is also subject-matter and experiment or project specific. In fact, S and R2 are just two different ways of describing the same thing: how well the statistical model fits the data. R2 is a proportion or percentage, while S is in the units of the response variable. S is the standard deviation of the residuals (model errors.) Since residuals from good models are normally distributed, the S value can be used to model the distribution of modeling errors in the same units as the response variable. This often makes it easier for subject-matter experts to tell you, the Black Belt, what an acceptable value of S should be.

One last thing, there are published papers that treat various statistical cutoff values, such as p-values, much more rigorously. For example, p-value cutoffs based on economic or risk considerations. If you have a deeper interest in the subject these papers are worth looking up. Look in journals such as The Journal of Quality Technology or Quality Engineering. Expect to see a bit of math in these papers.

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A Bulk Sampling Question

Tuesday, September 18th, 2012

Question from a Black Belt student:

The calculator sheet is very useful. But for my situation here in my plant, we produce bulk product (1200 MT of Refined Canola Oil). Suppose we need to test for “linoleic value” at the finished product, which is fluctuating between 1.75 and 2.39 values (margin of 0.64), sigma estimate is 0.26 and average of 2.06. So if we put this data in the sample size calculator we’d get a sample size of 1 (N unknown) or 0 (N known) at 0.05 Alpha risk. What is 1? 1 Metric Ton or 1 sample? And if it is one sample as I think it is, how large should be the sample? 4 Oz container or 1 Liter? Also if it is 1 MT that’s huge and expensive sample. Also I am confused about the zero number I got for the know N. How come that I do not meed sampling at all?

For your info, this is one of the main reasons I decided to pursue six sigma is to be able to identify sample size when I needed to. That has been a question in my mind since long time specially about bulk product and I feel good now that I am in touch with you so you can help me out on this.

My response:
Jack,

The sample size tools presented in the training are for what is known as “discrete sampling.” That is, for sampling discrete units such as people, automobiles, or other such distinct “widgets” that are separate entities. They can’t be used to calculate sample sizes for processes such as yours, which are referred to as bulk processes. That’s why your results make no sense.

Bulk process sampling is a unique application of statistical sampling. There are two primary bulk sampling questions: the science of testing and homogeneity. The science question asks what sample size is needed to obtain scientifically valid results. This is not a question of statistics per se, but one of science. It needs to be answered by subject matter experts. The question of homogeneity is also scientific, but it has statistical implications and statistics can help answer it. If the solution is perfectly homogeneous with respect to the property being measured, then all samples will produce the same result (except for measurement error, which is discussed at a later point in the training.) However, if the material is heterogeneous then you must construct a representative sample to properly characterize the lot of material. Here’s a good article on this topic. In this case the proper sample size will be whatever sample size is needed to characterize the lot. You may also find this article to be useful.

Another important topic is the sampling interval (as distinct from the size of a single sample.) If your canola oil is produced continuously rather than in discrete batches, you will want to look into this. However, from your description it sounds as if the canola oil is produced in discrete batches and I assume you’ll want to sample each batch using the procedure described above.

You mention that one of the main reasons you are pursuing a Lean Six Sigma Black Belt is to be able to identify sample size for bulk processes. I should point out that this is a specialized application of quality engineering rather than Six Sigma. While the Black Belt learns many of the tools of quality engineering, it doesn’t cover all of the QE body of knowledge. (The Black Belt also learns a number of soft skills that the QE doesn’t learn.) In fact, even traditional quality engineering training doesn’t cover the specialized topic of bulk process sampling, although it does go deeper into sampling than Black Belt training. What I’m saying is the you may want to supplement your Black Belt training with additional studies specific to your industry.

Tom Pyzdek

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Sigma Level Versus Process Capability Estimates

Thursday, August 23rd, 2012

From the Lean Six Sigma Green Belt Forum on 6-sigma-training.com:

Question

I am a little bit confused about the “process sigma level calculation” :

  1. The lesson presents a first method based on the RTY : First we have to compute the Normalized Yield and then derive the corresponding Z value. Finally we have to add 1.5 to account for long-term process variations and this is how we obtain the process sigma level.
  2. A second method is proposed based on Process Capability Analysis. More specifically, using Zu and ZL. These values are used to evaluate the overall defective rate so that we can extract the Z value. But this time, we are not adding 1.5 to get the sigma level ? Why ?

Thank you for your feedback,

Answer

It’s a difference in terminology: process capability versus process sigma. Process capability analysis has been around a lot longer than Six Sigma. It does not use the 1.5 sigma “fudge factor” used in process sigma calculations.  In the lesson on process capability analysis I note that “in Six Sigma we usually assume that the customer’s actual experience will be worse than what we predict in capability analysis. We add a fudge factor that assumes that the process will shift by approximately 1.5 standard deviations.” However, when calculating process capability, statistical software such as Minitab will just show what the capability is based on the actual data, i.e., the traditional way. If you wish you can convert Minitab’s PPM data to process sigma by using this calculator. This will include the 1.5 sigma shift.

Now let me confuse the issue even further. Let’s say that Minitab’s process capability analysis (normal) shows you the expected overall performance is a PPM of 50,000. I.e., 5%. If you look up Z for a 5% tail area in a normal table you will find it is approximately 1.645. However, if you enter 50,000 ppm into the sigma calculator it will return a process sigma value of 3.14 because it adds 1.5 sigma to the 1.645 and rounds the result down. One might ask why it does this. After all, it makes the process look better rather than worse.  The answer is that the 1.5 sigma adjustment should not be applied to ppm estimates from process capability analysis. The 1.5 sigma shift is meant to be applied to failures observed in the field to extrapolate back to what the PCA estimate originally was. For the above example, if we see a field failure rate of 5%, then the PCA at the production process probably was much better, 3.14 sigma.

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Sigma PPM Percent Converter

Friday, March 30th, 2012

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Why are Control Limits at 3 Sigma?

Tuesday, February 7th, 2012

A LinkedIn discussion started by Tham Nguyen Khoa asks:

Why [are] control limits on control chart are [sic] drawn at 3s?
Control limits on a control chart are commonly drawn at 3s from the center line because 3-sigma limits are a good balance point between two types of errors:

Type I or alpha errors occur when a point falls outside the control limits even though no special cause is operating. The result is a witch-hunt for special causes and adjustment of things here and there. The tampering usually distorts a stable process as well as wasting time and energy.
Type II or beta errors occur when you miss a special cause because the chart isn’t sensitive enough to detect it. In this case, you will go along unaware that the problem exists and thus unable to root it out.

Are there any more reasons?

The discussion goes on at great length (48 comments at the time this is written,) but I’ll just post my comment here:

Things like type I and type II errors apply to enumerative statistics. Control charts are analytic statistical tools, so these terms do not apply here. Type I and Type II errors can be stated with precision because, as enumerative statistics, inferences based on them apply to a static population. Analytic statistics, in contrast, are used to make inferences about the future performance of a dynamic process. Errors related to inferences about the future can never be precisely calculated.

That being said, the idea that tampering occurs when a process that is not being influenced by special causes of variation is changed as if it were, and that tampering makes matters worse, is certainly true. When we want to determine if a special cause is present in a process, we make use of data to help us decide. No matter what the data show, there is always a chance that we mistakenly conclude that a special cause exists (or doesn’t exist.) It’s obvious that the further a data point is from the “norm,” the smaller the probability that we’ll mistakenly conclude that a special cause is present. Shewhart did not base control limits on precise calculations of Type I or Type II error. He based them on the fact that in practice engineers at Western Electric were able to easily identify the special cause of variation when observations fell 3 or more sigma from the long term mean. They were more challenged to find a special cause for observations closer to the mean.

Think about it like this: if you created a list of everything that caused a process to change even a small amount you would have a very, very long list. You could never pin down the one big thing from this long list, because there is no one big thing. But if you ask for a list of everything that caused a process to change a lot, say by 3 sigma, that list would be relatively short. In between these two extremes are changes of intermediate magnitude and lists that vary between the long “any change list” and the short “3-sigma change list.” Just where to draw the line depends on a large number of things, such as the cost of checking out the possible causes on the list, the cost of missing something, the frequency that changes of a given magnitude occur, etc.. As a default starting point we can use 3-sigma to trigger our special cause search, if for no other reason than this has worked pretty well for 93 years. But that doesn’t mean that it should be accepted as dogma. What we are solving for are lines (control limits) that minimize total costs. In the end, it’s a management decision, hopefully one that’s based on facts and data.

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Where Do Those Six Sigma Statistics Come From?

Friday, January 13th, 2012

A student of mine had numerous questions about the various statistics used in Six Sigma. Here is my response to him in an open email:

The questions you are asking regarding “Where do these statistics come from?” require entire courses in statistics to answer. In Lean Six Sigma we take information from a dozen or so statistics courses, project management courses, psychology courses, business courses, mathematics courses, etc. and put it into an action framework that can be used to make fast improvements. We probably present less than 10% of the information you would receive if you sat through all of these courses, but we do so in less than 5% of the time it would take to complete all of these courses. It’s a tradeoff. We make the greatest compromises in the field of statistics. We discuss the use and interpretation of a select subset of statistics, and answer the question “where do these statistics come from?” by saying “they come from computer software.” While most are satisfied with this answer, some find the answer to be most unsatisfying. Judging from your questions, I suspect you are in the latter group.

anova-table-calculations-e-handbook-of-statistics

Two-Way ANOVA Calculations from E-Handbook of Statistics

Assuming you don’t have the time or the desire to take all of the courses relating to the Lean Six Sigma body of knowledge, but still seek answers to the specific statistics you asked about, I recommend the E-Handbook of Statistical Methods. This reference source is free and very comprehensive. It’s easy to search and will give you the answers you seek. For example, I searched on the term sum of squares, which you asked about, and the search returned pages on the half-normal probability plot (your question about fig. 10.26,) 1-way ANOVA (several of your question were about these calculations,) and several other related topics. A search on ss interaction provides answers to your question about the calculation of this intermediate statistic.

Sorry I can’t address all of your questions via email, but perhaps the reference above will start you on your way to answers. I had the same questions when I started learning about quality improvement nearly 45 years ago, and I am still looking for answers to questions today. Have fun!

Tom Pyzdek

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Statistical Surprises and Absurdities

Sunday, December 18th, 2011

I held a Webinar for Pyzdek Institute students entitled Statistical Surprises and Absurdities. Topics discussed included sampling bias, misused and misleading averages, distorting results by use of selective data weighting, selective reporting, missing information, distorted graphics, Say What? and So What? statistics, and much more! Here’s the recording

Here’s a link to the slides presented in the webinar.

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Pyzdek Institute Offers Free Statistics Course with Belt Training

Monday, November 28th, 2011

The Pyzdek Institute has announced that it is giving away a complete Statistics course with registration for any of its Six Sigma or Lean Six Sigma Green Belt or Black Belt training courses. The statistics course, which includes 4 DVDs and two follow-along printed guides, consists of 24 lectures of 30 minutes each. Part 1 (12 lectures) covers all of the subjects commonly included with college introductory statistics course. Part 2 (12 lectures) explores a wide variety of applications of statistical methods.These challenging yet accessible lectures assume no background in mathematics beyond basic algebra. While most introductory college statistics courses stress technical problem solving and plugging data into formulae, this course focuses on the logical foundations and underlying strategies of statistical reasoning, illustrated with plenty of examples. Professor Michael Starbird walks you through the most important equations, but his emphasis is on the role of statistics in daily life, giving you a broad overview of how statistical tools are employed in risk assessment, college admissions, drug testing, fraud investigation, and a host of other applications.

This offer is good only while supplies last. Click here to register or to get additional details.

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Free Webinar about a New Control Chart

Thursday, September 22nd, 2011

The Laney p’ Control Chart is an exciting innovation in statistical process control (SPC). The classic control charts for attributes data (p-charts, u-charts, etc.) are based on assumptions about the underlying distribution of their data (binomial or Poisson). Inherent in those assumptions is the further assumption that the “parameter” (mean) of the distribution is constant over time. In real applications, this is not always true (some days it rains and some days it does not). This is especially noticeable when the subgroup sizes are very large. Until now, the solution has been to treat the observations as variables in an individual’s chart. Unfortunately, this produces flat control limits even if the subgroup sizes vary. David B. Laney developed an innovative approach to this situation which has come to be known as the Laney p’ chart (p-prime chart.) It is a universal technique that is applicable whether the parameter is stable or not.

About Your Presenter, David B. Laney

David B. Laney

David B. Laney

David B. Laney worked for 33 years at BellSouth as Directory of Statistical Methodology. He is a pioneer at BellSouth in TQM, DOE, and Six Sigma. David’s p-prime chart is an innovation that is being used in a wide variety of areas. It is now included in many statistical applications, such as Minitab and SigmaXL. David is enjoying retirement with his family in the Birmingham, Alabama area.

Date: Wednesday, September 28, 2011

Session #1, 1:00 PM Eastern Time. Click here to register.
Session #2, 7:00 PM Eastern Time. Click here to register.

 

Update

Click here to view a video recording of David’s webinar.

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