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Explaining the Total Degrees of Freedom for Six Sigma Practitioners

George Woodley

George Woodley

We, as statisticians, Six Sigma Belts and quality practitioners have utilized the term degrees of freedom as a part of our hypothesis testing, such as the t-test for comparison of two means and ANOVA (Analysis of Variance), as well as confidence intervals, to mention a few references. I can recall from the many classes I have taught from Six Sigma Green Belts to Six Sigma Master Black Belts inclusive, that students have had a bit of a problem grasping the whole idea of the degrees of freedom, especially when we describe the concept of the standard deviation: …the average distance of the data from the MEAN…1 By now, Six Sigma practitioners should have a comfort level with concepts like the MEAN; which is calculated by taking the sum of all the observations, and dividing by the number of observations (n). The total degrees of freedom are then represented as (n-1).

Defining Degrees of Freedom

One method for describing the degrees of freedom, as per William Gosset, has been stated as, “The general idea: given that n pieces of data x1, x2, … xn, you use one ‘degree of freedom’ when you compute μ, leaving n-1 independent pieces of information.”2

This was reflected in the approach summarized by one of my former professors. He stated that the degrees of freedom represented the total number of observations minus the number of population parameters that are being estimated by a specific statistical test. Since we assume populations are infinite and cannot be easily accessed to generate parameters, we rely on samples to generate statistical inferences that provide estimates of the original population, provided the sampling techniques are both random and representative; another discussion for later.

This may seem very elementary, but from my own experiences, degrees of freedom have not been the easiest of concepts to comprehend—especially for the novice Six Sigma belt. A definition that can also be representative of the concept of the degrees of freedom can be summarized as “equal to the number of independent pieces of information concerning the variance.”3 For a random sample from a population, the number of degrees of freedom is equal to the sample size minus 1.

A Degrees of Freedom Numerical Example

A numerical example of this approach might illustrate this. The values would reflect the actual observations of a given data set. This example is simply for illustration purposes only. Given that we have eight data points that sum up to 60, we can randomly and independently assign values to seven of them. For instance, we may record them as: 8, 9, 7, 6, 7, 10 and 7. The seven values would have the freedom to be any number, yet the eighth number would have to be some fixed value to sum up to a total of 60 (in this case the value would have to be 6). Hence, the degrees of freedom are (n-1) or (8–1) = 7. There are seven numbers that can take on any value, but only one number will make the equation (the sum of the values, in this case) hold true.

One may argue that although this seems to be a simplistic illustration, the data collected for the original seven readings are not really independent, in that they are representative of an existing process, and depend on what the observations are in reality. Furthermore, we would have to know from the beginning what the final value was—in this case 60. Even though this illustration attempts to explain the theory behind the degrees of freedom it can be more confusing than obvious.

My “Easy” Way of Describing Degrees of Freedom

What really inspired me to write this article about the impact of the degrees of freedom was a conversation I had with my wife. She was heading to her class, and she called me and asked if I had an “easy” way of explaining the degrees of freedom. I gave her the description for describing the degrees of freedom I use in my classes:

Since statistics deals with making decisions in the world of uncertainty, we, as statisticians, need to provide ourselves with a cushion or padding to deal with this uncertainty. It can be viewed as the greater the sample size, the more confident we can be with our decisions. For example, when we estimate the variance of a normal distribution, we divide the sum of the squared deviations by (n-1). Hence if we have a sample size of 5, we are dividing by 4. This provides us with as a cushion of 20 percent. In fact, we are overstating the variance by a factor of 20 percent. If, however, our sample size is 100, we would be dividing by 99 percent. Here we are only overstating the variance by a factor of 1 percent.

This explanation places the emphasis on a common statistical concept that the larger the sample size, the more confident we can be of our estimates and decisions. To summarize this idea in a slightly different way—as long as our sampling technique is random and representative, the likelihood that we have a good estimator of a parameter increases as the sample sizes increase.

I have attempted to address the various approaches to the degrees of freedom and hopefully my simplistic approach to the rationale behind what we are trying to accomplish can shed some light on future explanations of such a vital part of statistical analysis.


1 Gonick, L. and Smith, W. (1993), The Cartoon Guide to Statistics, Harper Collins Publishers, pg. 22
2 Breyfogle, Forrest W. III (2003), Implementing Six Sigma, John Wiley & Sons, pg. 1105
3 Upton, Graham and Cook, Ian (2002), Dictionary of Statistics, Oxford University Press, pg. 100

©2009 E. George Woodley. All rights reserved.
Published with permission from the author.

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The Kano Model: Critical to Quality Characteristics and VOC

Origin of the Kano Model

Dr. Noriaki Kano, a very astute student of Dr. Ishikawa, developed an interesting model to address the various ways in which Six Sigma practitioners could prioritize customer needs. This becomes particularly important when trying to rank the customer’s wants and desires in a logical fashion.

The Practical Side to the Kano Model

The Kano model is a tool that can be used to prioritize the Critical to Quality characteristics, as defined by the Voice of the Customer, which I will explain in greater detail below. The three categories identified by the Kano model are:

  • Must Be: The quailty characteristics that must be present or the customer will go elsewhere.
  • Performance: The better we are at meeting these needs, the happier the customer is.
  • Delighter: Those qualities that the customer was not expecting but received as a bonus.

The First Step for Creating the Kano Model: Identifying the Voice of the Customer

The first step for creating the Kano model is to identify the quality characteristics that are typically fuzzy, vague and nebulous. These quality characteristics are referred to as the Voice of the Customer (VOC). Once the Voice of the Customer is understood, we can attempt to translate it into quantitative terms known as critical to quality (CTQ) characteristics. This should not be a new concept for those familiar with the Six Sigma methodology. What happens from here, though, can sometimes go astray if we are not careful and try to put our own spin on the needs of the customer. This may be the result of trying to make things more easily obtainable for us—a formula for failure.

Use the Kano Model to Prioritize the Critical to Quality Characteristics

So, now that we have identified what is important to the customer in workable terms, we can go to the second step. Always keeping the customer in mind, we can apply the concepts outlined in the Kano model diagram.

A Few Words About Kano

A Few Words About Kano

The Kano model is broken down into an (x, y) graph, where the x-axis of the Kano model represents how good we are at achieving the customer’s outcome(s), or CTQ’s. The y-axis of the Kano model records the customer’s level of satisfaction as a result of our level of achievement.

The red line on the Kano model represents the Must Bes. That is, whatever the quality characteristic is, it must be present; if the quality characteristic is not met, the customer will go elsewhere. The customer does not care if the product is wrapped in 24-carat gold, only that it is present and is functionally doing what it was designed to do. An example of this would be a client who checks into a hotel room expecting to find a bed, curtains and bathroom in the room. These items are not called out for by the customer, but would definitely cause them to go elsewhere if any of these “characteristics” were not present.

The blue line on the Kano model represents the Performance. This line reflects the Voice of the Customer. The better we are at meeting these needs, the happier the customer is. It is here where the trade-offs take place. Someone wanting good gas mileage would not likely expect to have a vehicle that has great accelerations from a standing position.

By far, the most interesting evaluation point of the Kano model is the Delighter (the green line). This represents those qualities that the customer was not expecting, but received as a bonus. A few years ago, it was customary that when a car was taken back to the dealer for a warranty oil change, the vehicle was returned to the owner with its body washed, mirrors polished, carpets vacuumed, etc. After a few trips to the dealer, this Delighter became a Must Be characteristic. Thus, a characteristic that once was exciting was now a basic need, and a part of the customer’s expectations. Another example of this is the amenities platter that some hotels provide their platinum customers upon checking in. I am one of those clients entitled to such a treat. This practice was certainly a delight. It has, however, become an expected part of my check-in, such that if there is no platter waiting in my room, I’m on the phone with the front desk.

Once the critical to quality characteristics have been prioritized, the last step of the Kano model involves an analysis of evaluating or assessing just how well we can satisfy each of Dr. Noriaki Kano’s classifications.

Kano Model Case Study

Being a trainer and consultant, I spend a lot of time on the road. In doing so, I have a tendency to check into hotels on a regular basis, as mentioned earlier. I once queried the manager of a hotel I spend a lot of time at on how he established practices to entice the business client. He related the following scenario to me.

The first thing he did was identify a list of qualities the client would be interested in. He came upon his list by listening to complaints, handing out surveys, holding focus groups and conducting interviews. The information below is a partial list from the Voice of the Customer. Knowing that I was involved in something that dealt with customer satisfaction, he asked me to assist him in ranking the characteristics. I explained the concepts behind the Kano model, and together we developed the list in the column labeled Business Client, as shown in Table 1. This was all fine and dandy, as far as the business customer was concerned.

Table 1

Table 1

For my own interest, I asked him to look at these same characteristics from the point of view of a vacationing family. As a final task, I asked him to assess how strong or weak he felt the hotel was when trying to meet those quality characteristics identified in table 1.

The results are shown in Table 2.

Table 2

Table 2

The conclusions from this effort can be as summarized by looking at the rows that have a characteristic in the Must Be category. With respect to the business client, this yielded express checkout, a comfortable bed, continental breakfast, internet hook-up and newspaper. The vacationer, on the other hand, had Must Bes that included price, comfortable bed, cable/HBO and a swimming pool.

Of these quality characteristics, the manager realized that the hotel was weak in the check-in and express checkout process, and internet hook-up. This Kano model exercise allowed the manager to better address the needs of the customer, based on their Critical to Quality characteristics. Now the work begins to minimize the gap of where the hotel is with respect to where the hotel wants to be.

One final thought: If a characteristic isn’t on the list, does that mean it can be ignored?

©2006 E. George Woodley. All rights reserved.
Published with permission from the author.

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