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

• Confidence intervals are graphical
• Confidence intervals are able to compare more than two hypotheses
• In any real-world situation the null hypothesis is always false. If your sample size is large enough you will always reject the null hypothesis. This often leads to positively silly behavior, such as many government regulations. Newspaper headlines are even sillier.
• The null hypothesis (literally, the hypothesis that there is no difference) is boring and uninteresting. Hypothesis testing promotes poor science by encouraging researchers to run one experiment and compare the results to this boring alternative. It would generally be better practice to develop and compare several hypotheses with each other.
• Confidence intervals are less confusing to students, lay persons, and quite frankly most statistics instructors

If you do some research you’ll find quite a body of literature complaining about the hypothesis testing approach. This is my small contribution to that cause.

Six Sigma is a discipline where many other disciplines are sampled. The idea is to provide a skill set to a cadre of change agents that can be used to complete practical projects in a relatively short period of time. These change agents are usually in this role for a limited time, after which they return to their “day jobs.” It has been proven beyond reasonable doubt that this yields amazing results in terms of successful change. But the Black Belt or Green Belt is, to a large extent, a Jack-of-all-Trades and the master of none. From time to time they will encounter the need for a true master of some particular skill. Enter the Master Black Belt. This person is first a successful Black Belt, but they are also masters of some specialized skill beyond the level normally acquired in Black Belt training.

My opinion is that Six Sigma Master Black Belts are “Six Sigma Black Belts, Plus.” That is to say, all good Master Black Belts first need to be experienced, successful Six Sigma Black Belts, usually for a period of two or more years. They must love the field and show a real knack for the Six Sigma Black Belt job. The plus is some additional area of expertise or skill set that is useful to the entire Six Sigma activity. Here’s a partial list of these skills, all of which I’ve seen used successfully by Master Black Belts at one or more client organizations.

• Six Sigma Master Black Belt-Statistics.  This person is an expert in statistics well above and beyond the statistical tools normally taught to Black Belts. Many have majored in statistics before becoming Black Belts and some are bona fide statisticians with Master’s Degrees. These are the folks Black Belts go to when those regression equations have the wrong signs on the coefficients or the problem has a mix of continuous and discrete independent variables and multiple dependent variables. They thrive on challenges involving the proper handling of complex analytical problems.
• Six Sigma Master Black Belt-Project Management. Project management, like statistics, is something all Black Belts learn in their training. However, the project management body of knowledge is large and most Black Belts learn only pieces of it. The Master Black Belt specializing in project management may also be a certified project manager. Or he or she may simply be adept at managing large projects, multi-generational projects, or meta-projects that are portfolios of related projects. Since the Master Black Belt position is not temporary, they can supervise projects that have a time frame that extends beyond the normal tenure of a Black Belt, thus assuring a smooth transition between project Black Belts.
• Six Sigma Master Black Belt-Trainer. Some people simply love to teach others and are gifted educators. Organizations who recognize this passion in a Black Belt will often take advantage of their good fortune to promote such individuals into a teaching role. As people with experience “in the trenches” these trainers are often well received by students because they can relate the subject matter to hands-on work as a Belt.
• Six Sigma Master Black Belt-Coach. Between the classroom and the successful delivery of results lies an immense chasm. Nearly all new Green Belts and Black Belts need the assistance of an understanding coach and mentor to help them bridge the gap. The coaching skill set is tough to find. Successful coaches are, of course, successful Black Belts. They must also possess a keen people sense, the patience of a monk, great teaching skills, and an intimate knowledge of the organization and how to get around obstacles. Their knowledge of statistics and project management, while not as developed as Masters specializing in these areas, must be well above that of the average Black Belt. If such a rare person can be found, the results can be amazing. A Master Black Belt Coach at one of my client companies coached a group of roughly two dozen Green Belts who delivered an average project benefit greater than that of the company’s Black Belts.
• Six Sigma Master Black Belt-Finance. Let’s face it, in many organizations the financial benefits reported for the Six Sigma activities are about as realistic as Harry Potter. Sooner or later the illusion will be shattered and the entire Six Sigma program will be in jeopardy. The importance of accurate accounting of Six Sigma results cannot be overemphasized. But often the attention given to this by the accounting and finance people is less than it should be. And it is often very challenging bringing a finance person with no Six Sigma background up to speed on a project. The Black Belt with finance and accounting skill can be a godsend in these situations.
• Six Sigma Master Black Belt-DfSS. Black Belts are taught the define-measure-analyze-improve-control (DMAIC) approach to improvement. DMAIC is used to improve and enhance existing products and processes. This is certainly useful considering that most organizations have massive opportunity in every aspect of their current operations. However, the future will always be where the very survival of the organization is determined. Brand new processes and products need to be developed if continued success is to be assured. Design for Six Sigma (DfSS) is the approach of choice for these opportunities. DfSS is a unique set of tools and techniques that are especially useful in the R&D process. Although Black Belts are taught a significant subset of these tools, the Master Black Belt-DfSS will learn many additional tools and a new framework for applying these tools. Sometimes the framework is very similar to DMAIC, such as DMADV (DMA are the same terms as in DMAIC, DV stands for Design and Verify.) But there are numerous DfSS approaches that bear little resemblance to DMAIC.

Of  course, the list could go on and on. However, this should be enough to give you an idea about my view of what Master Black Belts are, namely Black Belts with additional skills. These skills are often mixed and matched. For example, the Pyzdek Institute trains Master Black Belt-Trainer/Coaches to teach Six Sigma Belts our unique approach in addition to the complete Black Belt or Green Belt body of knowledge. But whether the Master Black Belt is the master of one or many specialized skill sets, they fill an important and often vital role in assuring the success of Six Sigma in their organization.

I recently attended a presentation using a control chart. The control chart showed a process in statistical control at about an 8-percent reject rate. The presenter noted that the process was stable and went on with her presentation. I barely avoided shouting that, while stability is nice, an 8-percent reject rate is not acceptable. The 8-percent level represents a certain amount of ignorance about the process; a level I find unacceptable. The problem is that the presenter didn’t think of it that way at all. To her, 8 percent represented a considerable accomplishment.

This blog is for those of you who, like me, want to scream that, as long as improvement is economically justified, “It’s never good enough!” I will present a way of measuring ignorance; a simple-to-compute statistic which highlights the fact that there is always something to learn about how to improve a given process.

First, let’s take a look at the philosophy that underlies statistics. In his book, The Art of Thinking, philosopher Leonard Peikoff wrote, “Statistics are applicable only when: 1. You are unavoidably ignorant about a given concrete; 2. Some action is necessary and cannot be deferred.”

In other words, if you’re trying to determine a course of action, your best bet is to acquire knowledge, not to blindly use statistics to guide you. While it’s true that we don’t want to tamper with a stable process, it’s also true that we don’t want to settle for anything other than the best level of quality we can provide. Control charts guide us away from tampering, but they don’t tell us how we can improve the process. Only new knowledge can do that.

Statistical probability should be used only when we lack knowledge of the situation and cannot obtain it at a reasonable cost. If we have direct knowledge about a situation, or can get it through a bit of research or by consulting someone who has it, then we should not blindly follow the statistical probabilities. In other words, if you know something about the situation, you should act on what you know.

Statistics are an expression of ignorance. They should only be used when ignorance is unavoidable, i.e., when knowledge is absent and unobtainable. Statistics are not knowledge. They are a calculation that permits action in the face of ignorance. This is the critical point missed by the presenter. She assumed that if she simply stated the level of ignorance, further improvement was not necessary.

Properly used, statistics measure ignorance or, conversely, knowledge. For example, assume that you want to buy a new piece of production machinery. Think of the important variables in the process as a list of 100 items, all of them unknown. You begin by creating a list of those items you believe to be important and prepare a plan to control as many of these items as possible. Let’s say you start with 75 items. Assuming that every item on your list is actually an important variable, these 75 items are special causes–things that affect your process and must be controlled. The remaining 25 items are common causes of variation, unknown to you but also important causes of process variability even though any one of these causes will have only a small effect.

From this starting point, you conduct a process capability study and, using statistics, quantify your knowledge as explaining all but +/-0.003″ of variation in the process. There are some out-of-control data points. After investigating these, you identify five more important variables. The process stabilizes, i.e., all of the remaining points on the control chart fall within the control limits.

Let’s assume that the control limits for the X-chart are now +/-0.002″. In philosophical terms, this means that you acquired +/-0.001″ of new knowledge, but +/-0.002″ of ignorance still remains. As time goes by and you learn more, the control limits will measure the amount of your learning. If in a year the control limits are at +/-0.001″, then you’ve learned enough to reduce the process variation by 50 percent.

As soon as you acquire this knowledge, the previous statistics become irrelevant. Gaining knowledge is the equivalent of converting special causes into common causes. This is like discovering more and more items on the list of things that cause your process to vary. You may never discover every item on the list, but with statistics to help you keep score, it’s fun to try. One way to make it even more fun is to plot a “knowledge chart.” Here’s how it works:

Record the process standard deviation from your most recent process control chart, for example, S₀ = 10.

For each subsequent complete control chart, compute the process standard deviation, for example, s₁ = 9.

Compute your relative knowledge,

k, as K=100% x (S₀-S₁)/S₀

For our example, K= 100% x (10-9)/10 = 10%

As you reduce your ignorance to zero, the knowledge measure will go to 100 percent. It’s a fun way to keep track of your quality progress!

“The numbers are the percentage of invoices sent out that contain errors.”  Joan explained.

Phil nodded his head and took a graphic out of his desk drawer and showed it to Joan (figure 1).  “Remember this?” He asked.

“Sure,”  Joan replied.  “It’s the decision tree for finding out which control chart to use.”

Control chart decision tree

Phil smiled and nodded.  He liked teaching and found it especially rewarding when bright students like Joan were able to apply what they had learned.

“But it doesn’t work for my data.”  Joan went on.

Phil’s smile faded.  “What do you mean it doesn’t work?  It has to work.”  He turned the paper towards her.  “Here, let me show you.”

He placed his finger on the dot at the left of the chart.  “Are we dealing with measurements or counts?”  He asked.

“Counts.”  Joan responded confidently.  “They count the number of invoices with errors.”

Phil ran his finger along the line labeled count until it reached the next dot.  “Okay, are we counting pieces or units, or are we counting occurrences?”

“Pieces or units.  They count invoices, which are pieces of paper.”

“And the last question is:  does the sample size stay the same, or does it vary?”

“It varies.  Every month a different number of invoices are processed.”  Joan said.

“Then the correct chart is the P chart.”  Phil said, tapping his finger on the graphic with an air of finality.

“Right.  That’s what I came up with.”  Joan said.  “But it doesn’t work.”

Phil looked perplexed.  “Of course it works.  Maybe you’re having trouble because it’s your first real control chart.  I’ll help you construct it.  Where are the data?”

Joan pointed to the sheet of paper she’d brought him.

“No, I need the raw data.  All this shows is the percentages computed from the raw data.”  Phil said.

Table 2

“That’s the problem, I don’t have it.”  Joan explained.  “All the sales offices send me are the percentages.  They don’t even keep the raw numbers.”

***

The problem is common.  The solution is, of course, to collect data in the proper format.  However, this may mean starting from scratch, changing habits firmly rooted in past behavior, expensive training, etc.  And it may take months to get enough data to establish valid control limits.  These obstacles are often enough to cause people to abandon control charts entirely.  Is there nothing Phil and Joan can do?

Fortunately, there is a solution: the X-chart, also known as the individuals chart or X and Moving Range chart.  Of course, the X-chart chart is recommended when plotting measurement data from “subgroups” of size 1.  But it is much more versatile.  The X-chart  is the Swiss Army Knife of control charts.  It can be shown that under the assumptions of statistical control and constant sample size, the central limit theorem can be used to show that the other control charts are mathematically related to the X-chart.  Better still, under conditions frequently encountered in practice, X-charts can be used to plot percentages, ratios, counts, and other non-measurement data even when the assumptions are only approximately met.

Real-World Example

Assume that Joan has the number of invoices and errors shown in Table 1.

Table 1

Figures 2 and 3 show, respectively, the p-chart and the x-chart for these data.  The x-chart was created by using the percentages as if they were measurements.  It is obvious that the conclusions are the same with either chart: the process is in statistical control.  Table 2 shows that the numbers calculated from the data are close too.

P-Chart

In other words, if all that are available are the percentages, the X-chart provides an excellent approximation to the P-chart.  The same conclusion applies to data for the np chart, c chart, u chart, sigma chart, etc.  In all of these cases, and more, the Swiss Army Knife of control charts gets the process operator focused on data. One could even argue that the simplicity of using a single chart instead of several charts outweighs the mathematical advantages in many cases.  So when in doubt, get the x-chart out.