Chapter 2 Measurements

Transcription

1 Chapter 2 Measurements 2.1 Measuring Things Measurement of things is a fundamental part of any scientifically based discipline. Some things are simple to measure, like the length of a piece of string or the time taken by a pedestrian to cross the road. Other things are very difficult to measure, like the size of an atom or the distance to Jupiter. Some things cannot be measured directly at all, like the volume of wood that might be harvested from a large forest area of thousands of hectares; there are simply too many trees in such a forest to measure them all and, as will be seen in Chaps. 5 and 6, it is quite difficult to measure the harvestable wood volume in even just one tree. When something is difficult to measure, or cannot be measured directly at all, methods of measurement are used to approximate or estimate it. These methods often involve measuring parts of the thing, parts which can be relatively easily measured. Then, more or less complicated mathematical procedures are used to convert the measurements of the parts to make an estimate of the size of the whole thing. Indeed, this book is concerned both with how parts of things in forests are measured, simple parts like the circumference of the stem or the height of a tree, and how those simple measurements are used to estimate a more difficult thing, like the harvestable wood volume in its entire stem. Whether a simple or very complex thing is being measured, there are three things about its measurement with which we should be concerned. These are the accuracy of the measurement, whether or not there is bias in it and what is its precision. The rest of this chapter will be concerned with these three issues, in the context of measurement of trees and forests. 2.2 Accuracy Accuracy is defined formally as the difference between a measurement or estimate of something and its true value. In simple terms, it can be thought of as how closely one is able to measure or estimate something, given the measuring equipment or P.W. West, Tree and Forest Measurement, 2nd edition, 5 DOI: / _2, Springer-Verlag Berlin Heidelberg 2009

2 6 2 Measurements estimation method available. Accuracy is expressed by saying that a measurement or estimate has been made to the nearest part of some unit of measurement, for example, to the nearest 1/10th of a metre, to the nearest hectare or to the nearest microsecond, depending on what type of thing is being measured. Suppose it was desired to measure something quite simple, like the length of the side of a field, of which the true length was 100 m. There are a variety of methods which could be used to do that. The simplest might be to simply pace the distance out yourself, having calibrated your paces by measuring their length along a tape measure. However, a result from pacing would not be expected to be very accurate, because a person is unable to keep each of his or her paces exactly the same length. Pacing would probably give a result for the length of the side of the field somewhere in the range of about m. That is, we could then say that measuring distances of around 100 m by pacing was accurate only to the nearest 5 m. A second method might be to use a measuring tape. Such tapes are often m long, made of fibre-glass, or other material which is not likely to stretch, and are usually calibrated in 1 cm units. Some care is needed with their use; they must be laid carefully along the ground and pulled tight to ensure that dips, hollows and irregularities in the ground surface influence the result as little as possible. However, even taking all due care with a tape like this, it would probably give a result for the length of the side of the field somewhere in the range m. That is, we would say the tape was accurate to the nearest 1/10th of a metre. A third method might involve a modern laser distance measuring device, such as used today by professional surveyors. Lasers are becoming very important for many types of measurement, not only in forestry ; their use in forestry is discussed further in Chaps. 4, 5 and 13. Laser is an acronym for Light A mplification by Stimulated Emission of R adiation. Laser light involves an intense, narrow beam of light of a single colour, which can be directed very precisely. The distance from an instrument to a solid object is determined by measuring the time it takes a pulse of laser light to be reflected from the object back to the instrument. These instruments contain very accurate clocks, capable of measuring the extremely short periods of time involved, given that light travels at about 300 million metres/second. A laser distance measuring device might be capable of measuring a distance of about 100 m with an accuracy at least to the nearest 1/1,000th of a metre, that is, to the nearest millimetre. The size of the thing being measured will immediately set some criterion for the accuracy required of the measurement. If one wishes to measure the sizes of atoms, which are of the order of 1 angstrom unit (Å) in diameter (an angstrom unit is one 100 millionth of a centimetre and was named after Anders Ångström, a Swedish physicist of the mid nineteenth century), complex laboratory equipment will be required, capable of taking measurements with an accuracy of fractions of an angstrom unit. If one wishes to measure the distance to Jupiter, which orbits the sun at an average distance of about 778 million km, a measurement method accurate to the nearest few tens of thousands of kilometres is probably what is required. However, the accuracy required ultimately of a measurement or estimate of something depends on the purpose for which the result is required. In turn, this will determine the sophistication of the equipment or estimation method required to achieve the desired accuracy.

3 2.3 Bias 7 Returning to the simple example of measurement of the length of the sides of a field, if it was desired to determine its area roughly, to work out how many bags of fertiliser were needed to cover it, the accuracy of measurement got from pacing out the sides would probably be adequate. On the other hand, if a professional surveyor wished to measure the field to establish the title to the property, a laser measuring device would probably be preferred to achieve the accuracy required by the legal system. 2.3 Bias Bias is defined as the difference between the average of a set of repeated measurements or estimates of something and its true value. In essence, if something is difficult to measure, it may not matter how many times we attempt to take the measurement, nor how many different types of measurement equipment we use, we may simply always get the wrong answer. By the wrong answer is meant that the results of the many attempts at measurement will be consistently larger or smaller than the true value of whatever it is that is being measured. If this is the case, the measurement or estimation method is said to be biased. By the same token, it would be said that the measurement or estimation method is unbiased if the average of the many measurement attempts differed negligibly from the true value. How small would the difference have to be to be considered negligible? Obviously, some limit is set by the accuracy of the measurement method; we simply cannot detect differences smaller than the accuracy. Apart from that, the degree of bias that will be considered acceptable will be determined entirely by the purposes for which the result of the measurement are to be used; this issue is discussed further in Sect To illustrate what is meant by bias, consider the problems involved in measuring the weight of the fine roots of a tree. Fine roots are the small (less than about 2 mm diameter), live roots at the extremities of the root system of a tree. Biologically, they are extremely important, because they take in the water and nutrients from the soil that the tree needs to survive and grow. Because of their importance, forest scientists need to measure them. The most appropriate way devised so far to do so is to excavate them from the soil. Obviously, this is a major task, since they will be scattered throughout a large volume of soil, extending perhaps 2 3 m or more away from the stem of a large tree and to a depth of 1 2 m. As well, so small and numerous are fine roots, it is very difficult to find all of them as one sorts laboriously through such a large volume of soil. Furthermore, in any patch of forest it is difficult to know if an excavated fine root belongs to the particular tree one is dealing with, or if it belongs to another, nearby tree or even to an understorey plant. So difficult are fine roots to find and measure, it is perhaps inevitable that that any attempt to do so is doomed to get the wrong answer, that is, to be a biased measurement method. Most probably, the answer will be an under-estimate of the true amount, because it is so difficult to find all the fine roots. There are various other methods used to measure fine roots (Sect ), all of them probably subject to bias, because of the difficulties associated with their measurement.

4 8 2 Measurements 2.4 Precision Precision is defined as the variation in a set of repeated measurements or estimates of something. The variation arises because of the limitations in the measurement or estimation technique, when it is used at different times and under varying circumstances, and limitations of the people taking the measurements. Following the example in Sect. 2.3, if a number of different people set out to measure the weight of the fine roots of a tree, it is inevitable that each would get a somewhat different result. So difficult are fine roots to measure, that individuals will vary in how many they manage to find in a large, excavated soil volume. Precision is measured by the amount of variation in the results of a repeated set of measurements of the same thing. The range of values in the set of estimates is one measure of precision. Another measure, called variance, is the measure used most commonly. Variance is a concept which derives from mathematical statistics. It is fundamental to a wide range of mathematical techniques used in science; these techniques deal with the problems that variation between natural things causes us in understanding how nature works. Variance and its use as a measure of precision will be discussed more fully in Chap. 9. Suppose the precision of a measurement technique is low, that is, a rather wide range of different results would be obtained when the technique is used by different people or at different times. If so, we would feel rather unsure about the extent to which we could rely on any one result we had obtained using the technique. In turn, we would not be very confident that we could draw worthwhile conclusions about whatever it was that was being measured. That is why precision is important in measurement. If it is high, we will feel confident that we can use the information to draw reliable conclusions. If it is low, we will feel much less confidence in our conclusions. 2.5 Bias, Precision and the Value of Measurements It is important to understand how bias and precision interact. This can be illustrated through an analogy used in various texts (Shiver and Borders 1996; Avery and Burkhart 2002), where a marksman is shooting at a target. In effect, the marksman is attempting to use a bullet to measure the position of the bullseye of the target. Figure 2.1 describes the analogy. The best possible result for the marksman is illustrated in Fig. 2.1(a). The average position of all the shots is right on the bullseye; that is, the average of the repeated attempts to measure the position of the bullseye does not differ appreciably from its true position, so it can be said to have been an unbiased measurement technique. As well, because the shots cluster closely around the bullseye, it can be said they measure its position with a high degree of certainty and so they represent measurements made with a high degree of precision.

5 2.5 Bias, Precision and the Value of Measurements 9 (a) (b) (c) (d) Fig. 2.1 Bullet holes in a target, as an analogy for bias and precision of measurements. ( a ) An unbiased, precise result, ( b ) a biased, precise result, ( c ) an unbiased, imprecise result and ( d ) a biased, imprecise result In the case of Fig. 2.1(b), the shots still cluster closely around one point, so they represent measurements made with a high degree of precision. However, their average position is some distance from the bullseye, so they represent a measurement technique in which there is bias. In this analogy, the bias might have arisen because the instrument being used (the gun) is not calibrated correctly, by having its sights set poorly. Or perhaps, unknown to the marksman, there was a wind blowing which pushed all the shots to the left. Figures 2.1(c) and (d) both show cases where the marksman has produced a wide spread of shots, which represent measurements made with a low degree of precision. In Fig. 2.1(c), despite their wide spread, the average position of the shots is still right on the bullseye, so they represent measurements made without bias. This might happen to a marksman on a day when the wind varies unpredictably, so that his or her shots are spread. Figure 2.1(d) represents the worst possible result for the marksman. Not only are the shots widespread, but also their average position is a long way from the bullseye. This might happen if the sights of the gun are not set correctly and if there are unpredictable wind variations. The important question then is whether or not a biased or imprecise measurement is still useful. Usually, it is better to have some measurement of something than no measure at all: what is difficult to judge is whether or not a biased but precise result (Fig. 2.1b ) is more useful than an unbiased but imprecise result (Fig. 2.1c ). Even more difficult to judge is if a biased and imprecise result (Fig. 2.1d ) is

6 10 2 Measurements better than no result at all. There are really no rules available to make these decisions. It becomes a matter of judgement for the person using the results to decide whether or not they are adequate for the purposes for which they are needed. As discussion of various measurement techniques continues throughout this book, reference will be made to the accuracy, bias and precision involved with them.

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