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

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

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About Your Presenter, 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.

But I still sometimes use other control charts.

The Problem

Organizations don’t do SPC for the fun of it. They do it because it helps them achieve their goals. Organizations exist to produce things of value for the benefit of customers, investors, and employees. They do this by transforming inputs into outputs of higher value via processes. They can do this better if they minimize variability of outcomes, which can best be accomplished by controlling the sources of variation in the inputs and processes. This is where SPC comes in. SPC is a methodology that uses statistical guidelines to help separate “special cause” and “common cause” variation. If a special cause of variation exists, it signals the need to act. Special cause variation is defined as a change of such a large magnitude that its cause can probably be identified if looked for at once. SPC operationally defines such a change as a measurement result more than 3 standard deviations from the process mean for whatever process metric is being monitored.

A problem might exist if the process generates measurements that are highly skewed, even when it is not being influenced by special causes of variations. Such processes are quite common in the real world. For example, nearly all measurements produced by geometric dimensioning and tolerancing are skewed, as are measurements of time-based phenomena such as those encountered in services industries including the healthcare and hospitality industries. Highly skewed distributions produce a relatively high percentage of results more than 3 standard deviations from the mean even if no special causes exist. In other words, they produce many “false alarms” that will trigger a search for a problem when there is no problem. The false alarms may even lead to tampering, thereby causing a stable process to become unstable.

I-Charts Don’t Solve the Problem

The skewed distribution problem is exacerbated by using I-charts. I-charts are relatively insensitive to moderate departures from normality, and very insensitive if the non-normality still produces a symmetric distribution. But for the data described above, this is not the case. If you use the I-chart for these data you will experience many false alarms. It’s just that simple.

The problem is to determine if a process is or is not being influenced by special causes of variation. A process distribution might appear as skewed because of special cause outliers, or because it naturally produces skewed data. The I-chart treats all data beyond 3 sigma as outliers; it doesn’t help you separate the natural, common cause process outcomes from special cause outcomes. Is the point beyond 3 sigma an outlying chicken, or a common cause egg? I.e., is the process being influenced by special causes, or only common causes? If the process data are naturally skewed you can’t answer this question using an I-chart.

A Simple Solution

The solution that I recommend is to begin your investigation with averages charts, also known as x-bar charts. Averages tend to have distributions that are approximately normal, even if the individual values are skewed. This means that, for a process with a skewed distribution that is not influenced by special causes, averages are much more likely to produce results that stay within 3 standard deviations of the mean than I-charts. It’s the best of both worlds: few false alarms, but still sensitive to special causes. If you have a nice run of subgroup averages without a special cause, plot a histogram of the data and see if the distribution looks skewed or symmetric. If the latter, you can use I-charts with confidence. If the former, stick with averages charts, or find a statistician or Master Black Belt to help you find a more advanced solution.

Stable Does Not Mean Normal

Before ending this article, I’d like to address another pet peeve of mine. I believe that too many teachers of SPC obsess on the need for normality. They confuse normality with the absence of special causes, also known as statistical control. I usually attribute this misunderstanding to a lack of experience with the real world, where normal distributions are so rare as to be virtually non-existent. By insisting on normality we encourage tampering and all of the problems associated with this approach to “process management.”

On the other hand, I am also impatient with people who insist that all non-normality be ignored. These individuals advocate using I-charts in all situations, regardless of the risk of false alarms. This attitude may also be due to a lack of experience. However, I’ve seen SPC lose its credibility when concerned process owners look for special causes over-and-over again without finding them. Like the boy who cried “Wolf!”, out-of-control signals become something to ignore. Eventually so does SPC.

My approach, which favors the I-chart but doesn’t make its use dogma, provides a rational middle ground.

Fig. 1-Large and Small Samples of Normally Distributed Data

One of the exercises I assign to students in my training involves creating two histograms from normally distributed random numbers. The results often look similar to those shown in figure 1. When I ask students to comment on their histograms I usually get comments about the averages, spread, and other statistical properties. However, that misses the point I’m trying to teach.

When we do Six Sigma we usually spend a lot of time mining historical data from databases. Sometimes the sample sizes are large, and sometimes they can be quite small. In fact, even large sample sizes can become small when we slice-and-dice them drilling down with various categories and sub-categories in search of CTQs. Statistical software will often automatically fit a normal curve to histograms created from these data. It’s often tempting to use the fitted curves to make an eyeball judgment about the normality of the data. Sometimes this is a good idea, and sometimes it isn’t. If the sample sizes are small, then the curve may not appear to fit the data very well simply because of small sample variation. Witness the top histogram in figure 1 for an example of a curve fitted to a histogram from a sample size of n=20. The histogram looks like a poor fit, but the p-value of a normality test tells us that the fit is pretty good anyway. So we’re probably safe assuming normality and acting accordingly.

The lower curve is fitted to a sample of n=500 data values. It appears to be a much better fit, and the p-value will back this conclusion. But what if the eyeballed curve fit and the p-value disagree?

Fig. 2-Decent Fit but Lousy P-value

Sometimes the fit of the curve is “close enough,” but the p-value will tell you that the fit is awful. Take a look at figure 2. The histogram suggests that the normal curve fits the data pretty well. There are many practical situations where you could use the normal distribution to make estimates and your estimates would be just fine. These are data on the time it takes to complete technical support calls. If you assume normality and you estimate costs or make a decision about process acceptability, your decisions will be essentially correct. However, the probability plot and AD goodness-of-fit statistic clearly show that the data are not normal and that the lack of fit is particularly poor in the tails (p < 0.005.) A closer examination shows that even in the tail areas the discrepancies are fractions of a percent. For example, the normal distribution estimates that 99.9% of all calls will take less than 35 minutes to complete, while the data show about 99.5%. Chances are these differences are of little or no practical importance.

The point is that in the business world we often need to make decisions and then get on to other, more urgent matters. The normal distribution is a handy device for getting quick estimates that are useful for such decisions. If your sample size is relatively large (say 200 or more) then you can go with the normality assumption if the fitted curve looks reasonably good. On the other hand, if you only have a small amount of data, you can still use the normality assumption if the histogram fit looks lousy, providing the p-value of the goodness-of-fit statistic says the normal curve is okay, i.e., if p > 0.05. The normality assumption is so useful that it’s worth using as a default, even if you bend the rules a bit.

QUESTION:
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Just a question that I thought I would run by you…
I work in the Automation industry, and am currently working on two production lines, and logging data for the parts being produced. One line is producing 60 parts per minute and I can thus log the data for every part. The other line is producing 240 parts per minute, and it is not possible to log the data for every part. I remember reading somewhere that in order to perform SPC you must take n consecutive samples (I think n was 5) every x number of cycles. What I need is definitive guidance on how to calculate n and x. I also need to know the statistical reason that n and x are used in order to explain this to the customer. Any feedback you can give me in relation to this would be greatly appreciated.

Regards,

AT in Irelend

RESPONSE:
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There is no rule that you need to sample n consecutive samples every x number of cycles. You are probably thinking of a technique known as PRE-Control, which is different than SPC. PRE-control also incorporates rules for deciding when to increase or decrease sampling frequency, stopping rules for processes, etc.. Personally, I don’t like PRE-Control for a variety of reasons, but if you have The Six Sigma Handbook, 2nd edition I discuss in starting on p. 661 or the 3rd edition starting on p. 465. My primary reason for disliking PRE-control is that it is a specification-based scheme (which I dislike in principle) and it will allow process variation to increase until it is as wide as the specs allow. SPC is all about reducing process variability to a minimum by identifying special causes of variation. When used in conjunction with Lean Six Sigma, SPC will also address common cause variation.

Instead of PRE-control I suggest that you consider using standard SPC control charts. I don’t know anything about your process so I can only offer general advice. If you’re already logging in metrics for 60 parts-per-minute I would be surprised if you’re not encountering problems like autocorrelation, which requires an adjustment to standard SPC such as using EWMA charts instead of classical control  charts. If you have autocorrelation and are not using the proper chart, then you will be experiencing a lot of “false alarms.” Processes seldom change by any meaningful amount in a matter of seconds, so you can probably extend the sampling interval. If you feel that you can economically sample 60 per minute, and that it is wise to do so, then you could sample this number of parts from the process running 240 parts per minute rather than checking every part. It would be best to choose the sample at random, rather than sampling every 4th part. Samples chosen using a fixed pattern are susceptible to problems if the process exhibits similar patterns. For example, if the process had 4 positions on a workstation then your 1-in-every-4 sample would always be sampling from the same workstation. Sometimes the patterns in the process are quite difficult to spot, and “Murphy’s Law” can strike at any time. Murphy’s Law states that anything that can go wrong, will go wrong.