Control Charts and the Central Limit Theorem

May 2017

(Note: all the previous SPC Knowledge Base in the control charts basics category are listed on the right-hand side. Select “SPC Knowledge Base” to go to the SPC Knowledge Base homepage Select this link for information on the SPC for Excel software.)

I first learned about control charts back in the early 1980s. Part of my initial training included being introduced to something called the Central Limit Theorem. I was taught that if the data were skewed, you needed to subgroup the data so that the subgroup averages were normally distributed. Then you could use the X-R control chart. This implied – and in fact was stated – that control charts work because of the central limit theorem. I was also taught that you should use the individuals control chart only if the data were normally distributed.

For some time, I believed the following was the approach to take. If I had skewed data, I would subgroup it – with little regard to how the subgroups were formed. Then I could place the data on aX-R control chart. I definitely made one mistake in doing this. That mistake was that my objective was to make a control chart for my process – not to control my process or to learn something about my process. There was no rational approach to the subgrouping I used. Just used enough data so the subgroup averages looked normally distributed.

So, it sounds like data needs to be normally distributed to go on a control chart. Right? And if you don’t have normally distributed data, then form subgroups so that the subgroup averages are normally distributed (the central limit theorem at work) so the control chart will work. Right? In addition, the three sigma limits are based on the normal distribution. Right? No, wrong in all three cases. But the shape of the distribution does make a difference in what out of control tests you may want to apply.

This month’s publication examines the relationship between control charts, the central limit theorem, three sigma limits, and the shape of the distribution.

You may download a pfd version of this publication at this link. Please fee free to leave a comment at the end of the publication.

In this issue:

• Introduction
• Central Limit Theorem
• Subgroup Averages and the Central Limit Theorem
• Subgroup Ranges and the Central Limit Theorem
• Three Sigma Limits
• Impact on Interpreting the Control Chart
• Summary
• Quick Links

Introduction

You never stop learning. That includes about control charts – something that has been around for years and years. I had my initial training and read many books on statistical process control (SPC). Then I ran across the works of Dr. Donald Wheeler (www.spcpress.com). I first saw Dr. Wheeler in one of Dr. W. Edwards Deming’s seminars around 1984. Dr. Wheeler was one of Dr. Deming’s masters – the ones we are supposed to learn from.

Over the years, the writings of Dr. Wheeler have given me many additional insights into the world of SPC. Dr. Wheeler is very clear that it is a myth that control charts work because of the central limit theorem. We will examine his reasoning on this. While he is correct about it being a myth, it does matter what the underlying distribution is when you apply out of control tests besides points beyond the control limits.

Central Limit Theorem

The central limit theorem can be stated as:

Regardless of the shape of the distribution, the distribution of average values (X) of subgroups of size n drawn from that population will tend towards a normal distribution as n becomes large.

The standard deviation of the distribution of subgroup averages (σ_X) is equal to the standard deviation of individual values (σ) divided by the square root of the subgroup size (n):

σ_X=σ/√n

We will use a randomly generated distribution to take a look at control charts and the central limit theorem. Our population is 1000 points randomly generated from an exponential distribution with a scale of 2. Figure 1 is a histogram of that distribution. We call this our population since it represents all possible outcomes from our process.

Figure 1: Histogram of 1000 Individual Data Points (Population)

You definitely would not consider Figure 1 to be normally distributed. Do we need to subgroup the data before we put these data on a control chart? Maybe and maybe not. It depends what you are trying to accomplish.

But we will begin by investigating the impact of subgrouping the data on the distribution of the subgrouped data, starting with subgroup averages.

Subgroup Averages and the Central Limit Theorem

Subgroup sizes of 2, 5 and 10 were used in this analysis. 500 subgroups were formed randomly from our population and the subgroup averages calculated. Figure 2 shows the distribution of subgroup averages when n = 2.

Figure 2: Subgroup Averages Histogram for n = 2

You can start to see the impact of the subgrouping. The histogram is not as wide and looks more “normally” distributed than Figure 1. Figure 3 shows the histogram for subgroup averages when n = 5.

Figure 3: Subgroup Averages Histogram for n = 5

Figure 3 is even more narrow and it looks more normally distributed. Figure 4 shows the histogram for the subgroup averages when n = 10.

Figure 4: Subgroup Averages Histogram for n = 10

Figures 2 through 4 show, that as n increases, the distribution become narrower and more bell-shaped -just as the central limit theorem says.

This knowledge is used in many statistical techniques. For example, if your subgroup size is large enough, you can reasonably assume that the subgroup averages are normally distributed and use the normal distribution to construct a confidence interval about a subgroup average.

What about charting the data from Figures 1 through 4 on a control chart? What would be the differences you see? Figure 1 is definitely skewed. There would be more points below the average on that control chart than on the control charts for the subgroup averages shown in Figures 2 to 4. Does it matter on a control chart if the data are not symmetrical about the average? What do you think? It does have an impact if you choose to use certain rules for out of control points as we will see.

But what about subgroup ranges? What impact does the central limit theorem have on these?

Subgroup Ranges and the Central Limit Theorem

When the subgroup averages were calculated as described above, the subgroup ranges were also calculated. The histograms for these are shown in Figures 5 to 7. What do you see as you look at these three figures?

Figure 5: Subgroup Ranges with n = 2

Figure 6: Subgroup Ranges with n = 5

Figure 7: Subgroup Ranges with n = 10

They don’t seem to be behaving like subgroup averages. They don’t seem to be forming a normal distribution that is tighter than the original data as the subgroup size increases. So, what does this mean to us? It means that the central limit theorem does not hold for subgroup ranges.

And this is the point that Dr. Wheeler makes: “If the central limit theorem was the foundation for control charts, then the range chart would not work.” Pure and simple. He has shown that it is a myth that control charts work because of the central limit theorem.

Part of the confusion comes it seems from how control limits are set. It is easy to see the familiar normal distribution with 99.73% of the data within ± three sigma of the average. The control limits sit right at ± three sigma marks. And how many times have we seen a normal curve superimposed on a control chart? So, it is easy to believe that data should be normally distributed to be put on a control chart. But that is not true. Three sigma limits were not chosen because the data were normally distributed.

Three Sigma Limits

Walter Shewhart, the father of SPC, used three sigma limits for control charts. But it was not because of the central limit theorem and the data being normally distributed. It was because that ± three sigma will contain the vast majority of the data regardless of the shape of the population’s distribution. And the three sigma limits represented a good economic trade-off when looking for special causes. The exponential distribution in Figure 1 only has 17 points out of 1000 beyond three standard deviations. Table 1 summaries that the percentage of points that are beyond the three-sigma limits for each of the histograms of subgroup averages above.

Table 1: % of Points Beyond the Three-Sigma Limits

Dr. Wheeler’s point is that there is a small chance of a false alarm – a control chart showing that there is a special cause because a point is beyond the control limits when it is actually due to common causes – if you use three sigma limits – regardless of the underlying distribution. But this applies to points beyond the control limits only. What about false alarms with the other out of control tests? Does the underlying distribution matter for the zones tests, for example?

Impact on Interpreting the Control Chart

The Western Electric rules involve dividing a control chart into zones and applying certain tests to those zones. These rules are covered in several of our earlier publications. For example, 2 out of 3 consecutive points in Zone A or beyond is an out of control situation. Or 8 points in a row on one side of the average is an out of control situation. Does the underlying distribution make a difference for these types of rules? The answer is yes, it does.

These tests are built around having a somewhat symmetrical distribution with about half the points above the average and half below the average. Figure 1 is not symmetrical. In fact, almost 63% of the points lie below the average. This means there is an increased likelihood that there will be more runs below the average than above the average. If your data are not somewhat symmetrical, be careful about applying the zone tests. Figure 8 shows the individuals chart (X) for the first 500 points from the exponential distribution shown in Figure 1. The red points represent out of control points from the zone tests. You can see how many there are below the average. These are not due to special causes but to the distribution of the data.

Figure 8: Individuals Control Chart

For an more in-depth discussion of control charts and non-normal data, please see our June 2014 publication.

Summary

This publication has examined the impact of the central limit theorem on control charts. It is myth that control charts work because of the central limit. Dr. Wheeler showed that very clearly since the central limit theorem does not work with the range chart. Part of the confusion is over the issue of three sigma limits. They were not chosen because of the normal distribution, but because, regardless of the distribution, most of the points lie within ± three standard deviations of the average. But the shape of the distribution does make a difference in what out of control tests you apply. You should use care in applying the zones tests if the data you are plotting is heavily skewed. You may get false alarms.