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Variation part 1: Distribution and the Central Limit Theorem

Quality Tools > Tools of the Trade > 62: Variation part 1: Distribution and the Central Limit Theorem


For the next several articles, we are going to go back to basics. Experienced people can thus thumb past the page or pass it on to a junior colleague. People newer to the quality profession, however, should pay close attention.


Natural and unnatural variation

Things vary. The weather changes each day. Cups of coffee taste good or not so good. Product quality is sometimes perfect and sometimes not so great. We may not be able to control the weather, but by understanding the nature of variation, we can create great cups of coffee every time and manufacture products that do not fail. Management of variation is thus a critical tool of many quality professionals.

One of the basic tenets of variation is that some things vary due to natural causes, whilst others vary due to unusual circumstances. Thus, looking at the shooting target in Figure 1, the hole off to the left is very likely due to something unusual (for example someone distracting the shooter at the point of firing), whilst the group of holes on the right are caused by things which occur every shot, including uncontrollable muscle movements, weak vision, varying bullet weight, etc. These are, respectively, special causes of variation and common causes of variation.



Fig. 1. Common and special causes


The way you should respond to these causes of variation differs significantly. Special causes can be dealt with on a one-by-one basis. If sudden distractions are causing stray shots, then you can keep people away. However, if you try to respond to common (or natural) causes one by one (often called tampering), then you are likely to make matters worse, as in Figure 2. Shot 1 is left of centre. Reacting to this by moving the gun to the right results in shot 2 being further to the right than it would otherwise. Likewise, shot 3 also becomes worse, and so on.




Fig. 2. The effects of tampering


Tampering is very common in many companies, as managers who do not understand the underlying process try to react to individual events. At best, no improvement is made. At worst, they break the process and seriously damage the company.


The only way to address common causes of variation is close and careful study of the process. In the shooting scenario, for example, bullets might be weighed and the shooter video-recorded from multiple angles to determine small variations in position and movement when firing.


Distribution of results

If the results of a set of measurements are plotted as in Figure 3, then some are closer to the centre and some are further out. The shape of this distribution is important, as if it is predictable, then you can make a proablistic forecast about future measurements and hence plan more accurately, assess future costs, etc.



Fig. 3. A Normal distribution


The most common shape to this curve is a bell-shape, which is commonly called a Normal or, sometimes, a Gaussian distribution. This is particularly useful as it has a mathmematical basis and can be used to predict and manage variation. Although the Normal distribution is very common, in practice other shapes of curve can be identified. The bottom line is always the same: if you can model the curve you can predict future measurement and hence control variation.


The Central Limit Theorem

A very interesting property of measurements, is that you can start off with any shape of distribution and force the measures into a Normal distribution. The Central Limit Theorem states that, even where the underlying population distribution is not Normal, the distribution of the averages of a set of samples will be approximately Normal.

This is clearly illustrated in Figure 4, which shows the distribution of average values achieved by throwing all possible combinations of one, two, three and four dice.

With a single die, the distribution is rectangular, as there is one, equally likely way of achieving each number. With two dice, the distribution becomes triangular, as although there is only one way of averaging one (two ones), there are six ways of averaging the central value of 3.5 (1-6, 2-5, 3-4, 4-3, 5-2 and 6-1).

With three dice, the distribution becomes curved, and with four dice it is markedly bell-shaped, as there is still only one way of averaging one, but there are four ways of averaging 1.25 (three 1s and a 2) and so on up to 147 ways of averaging 3.5!

A key use of this effect is that a predictable Normal distribution can be produced by measuring samples in groups of as few as four items at a time.



Fig. 4. The Central Limit Theorem in action


Next time: Variation part 2: measuring centre and spread


This article first appeared in Quality World, the journal of the Institute for Quality Assurance

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