2.3 Measurement Issues

It is pretty easy to get measurement data. However, it is relatively difficult to obtain accurate and precise measurement data. For every measurement activity, the errors of measurement must be carefully identified and controlled. From a basic engineering viewpoint, it will be impractical to obtain nearly perfect measurements in most cases. We only need to measure accurately enough so that the measurements do not induce functional problems in the task for which the measurements are necessary. To dress appropriately for today's weather, we only need to know the temperature outside (±2 to 3°C). For weather forecasting purposes, we might well need to know the temperature accurately (to within ±0.01°C). Very different measurement tools will be used to collect each of these measurements, a household thermometer being very much cheaper than the electronic version needed by the weather bureau.

2.3.1 Accuracy and Precision

There are two distinct issues that we must deal with in the measurement problem, both related to the fact that we will always be limited in our knowledge of the exact value that an attribute may have for each and every instantiation. Our measurements will only be as good as our measurement tools. Thus, the terms "accuracy" and "precision" are important to us. When we discuss the accuracy of a particular measurement, we will mean the closeness of that answer to the correct (and possibly unknowable) value. Precision, on the other hand, refers to the number of significant digits in our estimate for a population value. For example, the number 3.10473423 is a very precise estimate of the value of π; however, the number 3 is a much more accurate estimate. This very simple concept is lost on many practitioners of the black art of computer science. The literature in this area is replete with tables of very precise numbers. Little consideration, however, is given to the accuracy of these same values.

2.3.2 Measurement Error

There are many possible ways that our attempts to measure will be thwarted by Nature. It seems that there will always be a noise component in every signal provided by Nature. There are two classes of errors in measurement that we must understand: methodological errors and propagation errors. Methodological errors are a direct result of the noise component introduced by the method of collecting the measurements. Propagation errors are cumulative effects of error in measurement when accuracy suffers as result of repeated computation.

There are three distinct sources of methodological errors. First, there is the error that will be introduced by the measurement tool itself. The marks on an ordinary ruler, for example, are not exactly one millimeter apart, center to center. Further, the width of the marks may vary, causing us to misjudge proximity from one mark to another. Second, there are errors that are inherent in the method itself. Earlier, the value π was discussed. Any value of π that we use in a calculation will always represent an approximation.

A third source of methodological error is a modern one. Computers now perform all of our calculations. These computers can only store a fixed number of digits for any one calculation. If, for example, we have a computer that will store a 24-bit mantissa for floating point calculations, then this machine can precisely represent only nlog₂ 10 = 24 or n = 7 decimal digits. The circumstance may arise that we have a measurement that is precise to ten decimal digits; but as soon as we store this on our hypothetical computer, we will now only have a measurement with seven digits of precision. The imprecision here is due to a type of rounding, namely truncation. Sometimes we seek to capture a more accurate value through symmetric rounding by adding the decimal value 5 to the least significant decimal digit before the truncation.

Measurement without error will occur only in an ideal world. As mortals, we do not live in this ideal world. Thus, when we perform measurements, we must understand and want to know just exactly how much error there will be. We see that our measurement x* is the sum of two terms, as follows: x* = x + ε, where x is the actual value of the variable as it exists in nature, and ε is the sum of the error components or the absolute error. Many times, it will be useful to talk about the relative error introduced by the measurements. This relative error is ε/x*. For our purposes, the relative error will be a more useful index, in that it shows us how far we have actually deviated from the actual or true value. A relative error of 1 in 100 is far worse than a relative error of 1 in 10,000. From another perspective, if we had an absolute error of measurement of 0.01, then for measurement on the order of 100,000 this would not be a problem. If, however, our measurements were on the order of 0.001, an absolute error of 0.01 would be devastating.

It is also possible to characterize relative error in terms of x if we are assured that ε is substantially less than x, as follows:

Observe that:

Then:

which is to say that ε/x* ≈ ε/x when the relative error squared is negligible.

It is most important to understand that the measurement is an estimate of a value known only to Nature. We must factor this into our thinking whenever we engage in the act of measurement.