VariationVariation is a major cause of quality problems and consequently many process improvement activities focus on identifying and reducing it. It is thus important to understand as much about variation as possible, even though its statistical nature can be disconcerting.This chapter discusses the basic principles of variation, keeping the mathematical content to a minimum. In keeping with the style of the book, calculations are also minimized, consisting only of inserting numbers into simple formulae and using illustrated panels for calculations involving lists of numbers.There are four main sections in this chapter: Understanding variation: Key principles of this important topic. Measuring variation: Principles of measurement of variation. Measuring centering: Mean, median and mode. Measuring spread: Range and standard deviation.Understanding variationThe continuously variable nature of the universe is at the heart of the science of statistics, and at first glance can look very complex, particularly if approached from a mathematical viewpoint. This can lead to it being ignored, which is a pity, as even a simple appreciation of it can result in a reduction in haphazard attempts to control it, with a consequent saving in wasted time and degraded performance.What is variation?When a process is executed repeatedly, its outputs are seldom identical. For example, when a gun is successively fired at a target, as in Fig. 1, the bullets will not all pass through the same hole.

Fig. 1. Variation in targeted resultsThis lack of repeatability is caused by thevariationorvariabilityin the process. If these causes are understood, then this can lead to the development of solutions to reduce the variation in the process and result in more consistent products which require less inspection and testing, have less rejection and failure, cost less to build, have more satisfied customers and are more profitable.

Causes of variationVariation in process output is caused by variations within the process. These may be one or more of:1. Differing actions within the process.2. Differing effects within the process.3. Differing inputs to the process.As an example for each of these conditions, the variation in the placement of the bullet holes in the target may be affected by:1. The gun being held or used differently.2. Wear in the hammer mechanism causing the shell to be struck differently.3. The bullets being of slightly differing shape or weight.Thus, even if the first point is eliminated by putting the gun in a clamp and firing it remotely, the bullets will still not all hit the target in the identical position.The reasons why variation occurs can be divided into two important classes, known as common and special causes of variation. These are discussed further below.Common causes of variationWithin any process there are many variable factors, as indicated above, each of which may vary a small amount and in a predictable way, but when taken together result in a degree of randomness in the output, as indicated in the figure below. These seemingly uncontrollable factors are calledcommon causes of variation.Common causes of variation can seldom be eliminated by 'tampering' with the process. For example, consider the effect of simple adjustments to the clamped gun, as in the figure below.

Fig. 2. Tampering1. The first hole is to the left of center, so the clamp is rotated a little to the right.2. If the clamp had been left alone, the second bullet would have gone a little to the right of center, but as it has been moved right, the bullet now goes further to the right. As a reaction to this, the clamp is rotated somewhat more to the left.3. The third bullet tends towards the left anyway, so the result is a hole even further to the left.It can be seen from this that it would have been better not to tinker with the clamp, and that the score would be more likely to improve if the whole system were understood first and then fundamental improvements made, such as building a better gun or making better bullets.Special causes of variationSpecial causes of variation are unusual occurrences which come from outside the normal common causes, for example where a shot goes outside the main grouping, due to someone tripping over the gunner as the gun is fired, as below:

Fig. 3. Special and common causes of variationSpecial causes can thus be addressed as individual cases, finding the cause for each occurrence outside the normal grouping and preventing it from recurring. This may be contrasted with the way that common causes must be addressed through the overall process.The way that causes are addressed in a process improvement project is usually first to recognize and eliminate special causes, and then to find ways of improving the overall process in order to reduce common causes of variation.Static and dynamic variationThe distribution of measurements as described above takes no account of time or sequence, as it is not important which measurement came first or last. This isstaticvariation.If the order in which measurements are made is known, then significant trends may be detected, which may be useful for catching a problem before it becomes serious. This isdynamicvariation.For example, if the gunner is initially accurate, but becomes less so as his arm tires, then this may not be detected from the final positioning of holes on the target - it could only be seen by plotting the positioning of the holes across time.Dynamic variation is commonly measured using the Control Chart.Measuring variationVariation is not simple to measure, as by its nature is random and individual events cannot be predicted. Despite this, a degree of measurement can be achieved by looking at how a number of measurements group together. Usually these items are selected with sampling methods.The spread of measurements within a group enables special causes of variation to be distinguished from common causes of variation. Beyond this, the characteristics of how these random events are spread out can allow improvements in seemingly random chaos to be simply measured.Distribution of resultsIt is common in processes for most measurements to cluster around a central value, with less and less measurements occurring further away from this center. For example, the distribution of holes across the target will gradually spread out from a central, most common placement, as below:

The Normal distributionThe bell-shaped curve in the figure above occurs surprisingly often and is consequently called aNormal distribution(orGaussiandistribution, after its discoverer) and has some very useful properties which can be used to help variation be understood and controlled.Other distributionsA Normal distribution of measurement values does not always occur, and other distributions may be caused by various factors, conditions and combinations. Several of these are discussed in Chapter 23. It is a trap to use tools that expect a Normal distribution, such as Process Capability, when the distribution is not Normal.The Central Limit TheoremThe reason for the common occurrence of this Normal distribution is either a natural distribution or the very useful and remarkable effect described by theCentral Limit Theorem. This states that, even where the underlying population distribution is not normal, the distribution of the averages of a set of sampleswillbe approximately normal.This is clearly illustrated below, 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.Measuring distributionThe measurements of a process can vary in two different ways, in terms of theircenteringand theirspread, as illustrated below:

The centering (also calledaccuracyorcentral tendency) of a process, is the degree to which measurements gather around a target value. The spread (also calleddispersionorprecision) of the process is the degree of scatter of its output values.Measuring centeringTo measure the centering of a process requires that the center point of the set of results be identified. The accuracy of the process can then be determined by comparing it with target values. There are three ways of measuring this center point: the mean (or average), the median and the mode (see the figure below).

Fig. 1. Mean, median and mode in distributionsMeanThe most common way of measuring the center point of a set of measurements is with the average, ormean(i.e. the sum of all measurements divided by the total number of measurements).The mean is useful for further mathematical treatment, as it considers all values (although a few extreme values can cause the mean to become unrepresentative of the rest of the values).MedianIf the measurements are listed in numeric order, then themedianis the number half-way down the list. If there is an even number of measurements, it is half-way between the middle two numbers. The median is not distorted by extreme values, but it can be very unrepresentative of the other values, particularly in a distribution which is not symmetrical.ModeThemodeis the most commonly occurring measurement. In a distribution graph, this is the highest point. The mode is also not distorted by extreme values, and is useful for measuring such as average earnings. However, there can be more than one mode, and it is not as good as the mean for mathematical treatment.In a symmetrical distribution such as a Normal distribution, these three measures are the same. In anasymmetrical(orskewed) distribution, as below, there is a simple rule-of-thumb formula which can be used to estimate one, given the other two:Mean - Mode = 3 x (Mean - Median)Measuring spreadThere are two main ways of measuring the degree of spread of a set of measurements: the range and the standard deviation.RangeThe range of a set of measures is simply the difference between the largest and the smallest measurement value.Thus, for example, if you have a set of measures (21, 22, 26, 19, 12, 24, 33) then you first find the highest measure (33) and subtract the lowest measure (12) to give the range (21).This is easy to calculate, but there can be several problems with using it: Special causes of variation can cause an unrealistically wide range. As more measurements are made, it will tend to increase. It gives no indication of the data between its values.

Standard deviationThe standard deviation is a number which is calculated using a simple mathematical trick (calculating the square root of the average of squares) to find an 'average' number for the distance of the majority of measures from the mean.The standard deviation is of particular value when used with the Normal distribution, where known proportions of the measurements fall within one, two and three standard deviations of the mean, as below.

Fig. 1. Percentages in Normal Distribution between Standard DeviationsThus, given a set of measures, the mean and the standard deviation can be calculated, and from this can be derived the probability of future measures falling into the three bands, provided that the distribution is normal (a simple visual test for this is to draw a histogram and look for the bell shape).For example, if the gunner has an average score of 56 per target card, with a standard deviation of 6, then, provided the distribution is normal: 68.3% of scores will be 56 6 (= between 50 and 62) 95.4% of scores will be 56 12 (= between 44 and 68) 99.7% of scores will be 56 18 (= between 38 and 74)or, breaking out the six bands: 2.1% of scores will be between 38 and 44 13.6% of scores will be between 44 and 50 34.1% of scores will be between 50 and 56 34.1% of scores will be between 56 and 62 13.6% of scores will be between 62 and 68 2.1% of scores will be between 68 and 74 The remaining 0.3% of scores will be below 38 or above 74.

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