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.

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

I‘m often asked about the term “black belt” as it relates to six sigma. What, precisely, is a black belt? Where did the term originate? For that matter, where did the term “six sigma” originate? And, while we’re on the subject, what’s a green belt or master black belt?

Let’s start with the term “six sigma.” In a conversation with Ed Bales of Motorola University, I learned that Motorola coined the term in 1986. As those who have worked in quality for a while know, this term has statistical roots in the technique known as process capability analysis. Prior to the Japanese industrial invasion of U.S. markets, quality practitioners were happy with three sigma quality, which translates to about three errors or defects per 1,000 items for processes in a state of statistical control. Motorola discovered that its processes weren’t in statistical control–estimates based on field failure data indicated that Motorola’s processes apparently drifted by an average of 1.5 standard deviations. In a conversation with ex-Motorola trainer Mikel Harry, I learned that he considers the Cpk index–which measures short-term process variability under statistical control–worthless. Harry prefers the Ppk index, which measures actual performance rather than process capability. (Note that many experts, including me, disagree strongly with Harry on this issue.) In any case, before computing expected process failures, Motorola adds this 1.5 standard deviation. Thus, when we hear that a six sigma process will produce 3.4 parts-per-million (PPM) failures, we find that this PPM corresponds to the area in the tail beyond 4.5 standard deviations above the mean for a normal distribution.

Motorola also adopted the terms “black belt” and “green belt.” For my book The Six Sigma Handbook, I did extensive research into what employers expect of people with these titles. Here is a summary of these various responsibilities:

• Master black belt–This is the highest level of technical and organizational proficiency. Because master black belts train black belts, they must know everything the black belts know, as well as understand the mathematical theory on which the statistical methods are based. Masters must be able to assist black belts in applying the methods correctly in unusual situations. Whenever possible, statistical training should be conducted only by master black belts. If it’s necessary for black belts and green belts to provide training, they should only do so under the guidance of master black belts. Because of the nature of the master’s duties, communications and teaching skills should be judged as important as technical competence in selecting candidates.

• Black belt–Candidates for technical leader (black belt) status are technically oriented individuals held in high regard by their peers. They should be actively involved in the organizational change and development process. Candidates may come from a wide range of disciplines and need not be formally trained statisticians or engineers. However, because they are expected to master a wide variety of technical tools in a relatively short period of time, technical leader candidates will probably possess a background in college-level mathematics, the basic tool of quantitative analysis. College-level course work in statistical methods should be a prerequisite.

Six sigma technical leaders work to extract actionable knowledge from an organization’s information warehouse. Successful candidates should understand one or more operating systems, spreadsheets, database managers, presentation programs and word processors. As part of their training they will be required to become proficient in the use of one or more advanced statistical analysis software packages.

• Green belt –Green belts are six sigma team leaders capable of forming and facilitating six sigma teams and managing six sigma projects from concept to completion. Typically, green-belt training consists of five days of classroom training and is conducted in conjunction with six sigma team projects. Training covers facilitation techniques and meeting management, project management, quality management tools, quality control tools, problem solving, and exploratory data analysis. Usually, six sigma black belts help green belts choose their projects prior to the training, attend training with their green belts and assist them with their projects after the training.

Although the martial arts terms described above are common, they are by no means universal. Companies and consulting firms often create their own titles to describe the work done by these technical leaders.

Imagine the following scene. The boss rushes into the quality director’s office. He’s obviously distraught.

(Boss enters, walking quickly from stage right.)

Boss: “Jane, we’ve got a serious problem. Our biggest customer just called. Their assembly line is shut down because the last batch of XYZ-50′s that we shipped won’t fit into their assembly fixtures. What happened?”

(Jane, sitting at her desk, puts down her pen and looks up at her boss. She shakes her head in dismay.)

Jane: “I knew this would happen sooner or later, boss. The problem is that our customer requires us to provide a Cpk of 1.33 or higher. But the formula they make us use assumes normality, and the XYZ-50 has a skewed distribution. If we center the process to maximize Cpk, then the tail area extends beyond the specification limit .”

(Boss exits, stage right, shaking his head and wearing a puzzled expression.)

I fear that when the quality profession talks about process capability, this is how we sound to others. To many quality engineers and managers, process capability is a jumbled confusion of ideas expressed in jargon that only the anointed can understand. Let me try to clear the air on the subject.

Process capability is about one thing, and one thing only: quality. It answers the simple question, “Can you meet my requirements?” Ideally the customer would like a simple answer, yes or no. Unfortunately, this is not possible due to one or more of the following:

Inspection is not perfect; even 100-percent inspection won’t guarantee 100-percent quality. Explaining this becomes complicated.

All processes vary, and the variation must be analyzed using statistical methods that always predict at least an occasional failure. The statistics virtually always get complicated.

Measurement isn’t perfect, so even if a process did have zero variation, our measurements would still vary. This means that we might accidentally ship a defective item even if we measure it carefully. Not only that, our measurements of a particular item might be somewhat different from our customer’s measurements. Explaining how two trained people using the same type of instruments can check the same item and get different results can get complicated.

We or our customer might not properly understand the requirements. Human communication is always complicated.

Yet it’s really not complicated at all. In fact, the customer’s question can be answered easily, and the answer is: no. For all of the reasons listed, and many more, we cannot guarantee that we will always deliver a product or service that meets the customer’s requirements as understood by the customer.

So, now what? The best approach is also the most radical: Be honest. Tell customers about how many items they are likely to receive, on average, that will not meet the requirements. This cuts right to the heart of the matter. It tells customers what they want to know. It works for variables data and attributes data. If control charts are being used, the estimate can be obtained directly from the process average (for attributes data) or the process average and standard deviation (for variables data). The count can be adjusted to include sorting operations, inspection error, measurement error and all of the other factors that influence what the customer receives.

If our process is extremely good, we can tell the customer that, while we can’t guarantee perfection, we can provide quality in the near-perfect range. One good way of quantifying this is to use parts-per-million quality statements. For example: “Our return rate on this item is three returns per million items in service per year.” Most people can easily understand this statement. A customer ordering up to several thousand items will probably, and accurately, interpret this to mean “zero defects.”

If our process is less capable, stating the expected number of defective items that the customer will receive might result in a shock to both the employees and the customer. This may provide the incentive needed to improve quality.

High-volume production is another area where stating process capability as expected defectives can provide insights. A defect rate of 1/10 percent sounds pretty good. But a can line may produce in excess of 1,000 cans per minute, so a reject rate of 1/10 percent would result in the production of 1,440 defective cans per day. If the defect is major, say a leaking can that could damage many cases of product in a warehouse or truck, even a defective rate of one in a million might not be acceptable; it would result in several serious problems each week. For such processes, parts per billion quality may be required.

If a process is not in statistical control for unknown reasons, there is no way to state the process capability with any degree of precision. The best option is to tell the customer what the expected defectives will be (based on the historical data) and hope for the best.

The key to good customer relations is clear communication. The easiest way to get the point across is to tell the customer what level of product or service quality to expect, using plain language.

SPC benefits users by directing attention toward the facts and thus promoting reason and rationality in problem solving. In this posting, I’ll put that belief to the test by using SPC to explore an issue that has generated much emotion lately: global warming. Global warming is complex, dynamic, important and imperfectly understood. Statistical methods are designed to help us analyze just such processes.

The figure of merit in this case is the Earth’s mean temperature. Figure 1 presents a run chart of the data. The numbers are coded and show the deviation in average global temperature in hundredths of degrees C from the base period mean temperature. The base period is from 1951 to 1980. A value of 0 indicates an annual global mean temperature equal to the base period mean, while a value of 20 indicates a temperature 0.20° C below the average during the base period. The chart includes data from 1866 to 1996. (There is a comment about more recent data.)

Putting the data into a run chart shows 131 years of temperature variation at a glance. Initially, temperatures are cooler, roughly 0.50° C below the base period mean. At the end they are warmer, roughly 0.25° C above the base period mean.

In SPC, the preferred approach to determine potential long-term process performance is to conduct a process capability analysis. In a PCA, changes are carefully controlled to determine how the process behaves under ideal conditions. Control limits are computed from the PCA data and used to identify important changes that occur in the future.

Needless to say, we can’t to do this with many of our processes, including the global warming process. Instead, we are forced to deal with things the way they are. A first step is to compute the control limits for these data. To do this, we first must estimate the process average and standard deviation, s. The temptation is to compute the average and s by using a spreadsheet such as Excel, which gives us an average of 10.4 and s = 24.30.

However, computing s in this way only works if the process is in a state of statistical control. When the process’s state is unknown, it’s far better to base our sigma estimate on the “moving range.” A recent article in Quality Engineering shows that s can be estimated accurately by multiplying the median moving range by 1.047. With this method, we get s = 10.47.

The control limits are set at plus-and-minus three standard deviations from the long-term mean, giving 41.8 and 21.1 using coded measurement units. Figure 2 shows the control chart with the average and control limits drawn in. There are points below the lower control limit at the beginning of the chart, followed by points above the upper control limit at the end of the chart. This is an SPC definition of a trend.

We’ve now established that between 1866 and 1996, the global mean temperature measurements increased. If we compare the first 20 years on the chart to the last 20, the change is +64.4, or an increase of 0.64° C. The next step is to identify the special cause or causes behind this change. We will explore this issue in a future posting.