Abstract
This chapter aims to outline the technical and cultural contexts in which measurement systems, as presented in the previous chapter, are designed, set up, and operated. It first introduces the basic proposal that a measurement should produce as result not only one or more values of the property under consideration but also some information on the quality of those values, and discusses the consequences in terms of measurement uncertainty. This proposal is then embedded in the broader context of metrological systems, which help justify the societal significance of measurement results via their traceability to conventionally determined measurement units, so that measurement results can be interpreted in the same way by different persons in different places and times. Finally, we consider the issue of what is measured, i.e., the property of an object, or measurand, which must be somehow defined and identified, an act that is not necessarily related to assigning measurement values. On this basis, the chapters that follow develop and bring further specificity to our analysis and proposals. As with the previous chapter, we believe that the contents of this chapter should be sufficiently uncontroversial to be read and accepted by most, if not all, researchers and practitioners.
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Notes
- 1.
Eran Tal (2019) builds upon this distinction and argues that “due to the possibility of systematic error, the choice between [these two situations] is underdetermined in principle by any possible evidence”: we do not further develop his argument here. Moreover, there is a third case: whether the individual property did change or not, what may change over time is the definition of the general property of which the individual property is an instance, thus possibly making the measuring instrument inadequate. While this is (now) unusual for physical properties, this situation is (still) not uncommon in the human sciences, as for example for nursing ability, the very definition of which depends on the cultural context and therefore changes over places and times. We discuss the problem of the existence and identification of general properties in Sect. 6.6.
- 2.
In practice, this is possibly one of the sharpest distinctions between measurement in scientific and nonscientific contexts. Even the VIM admits that sometimes “the measurement result may be expressed as a single measured quantity value” and then acknowledges that “in many fields, this is the common way of expressing a measurement result” (JCGM, 2012: 2.9, Note 2).
- 3.
Like most of the contents of this chapter, what follows generally applies to both quantitative and nonquantitative properties, even though the mathematical aspects are mainly introduced here in reference to quantities. The issue of uncertainty in nonquantitative evaluations is further considered in Chap. 6.
- 4.
In addition, this concept of sensitivity should be distinguished from the analogous term used in biostatistics; the same comment applies to selectivity.
- 5.
The definition in the VIM (JCGM, 2012: 4.13) takes into account all influence properties at the same time, and therefore remains qualitative. Moreover, the VIM itself notes (4.13, Note 1) that in some contexts selectivity is considered as insensitivity to influence properties of the same kind as the measurand.
- 6.
The VIM definition is very general: “property of a measuring instrument, whereby its metrological properties remain constant in time” (JCGM, 2012: 4.19).
- 7.
When a measuring instrument is used in measurement, there are no known reference values to be applicable, as trueness should be evaluated with respect to the (unknown, if even existing) true value of the measurand. Hence the trueness of measurement results “is not a quantity” (JCGM, 2012: 2.14, Note 1).
- 8.
What is commented above about the trueness of measurement results also applies to accuracy: the accuracy of measurement results “is not a quantity” (JCGM, 2012: 2.13, Note 1).
- 9.
The uncertainty that is being addressed in this section (and elsewhere in this and other chapters) is not associated with sampling variability that is with the uncertainty that is due to a situation where a statistical result is based on a sample from a population of properties of distinct objects, where a parameter of the statistical distribution is being estimated—this is usually denoted as sampling error.
- 10.
This may be considered a measurement-specific case of the fundamental distinction between models and modeled entities, sometimes presented in terms of maps versus territory: the only “perfect” map is the territory itself, so that, paradoxically, aiming at a perfect map makes the mapping process useless (the subject of On Exactitude in Science, a delightful short story by Jorge Luis Borges). Analogously, a claimed-to-be-perfect measurement would directly exhibit the property under measurement (the perfect representative of itself, indeed), thus making the process of measuring pointless.
- 11.
The terms chosen in the VIM are even more explicit: “Type A evaluation of measurement uncertainty” and “Type B evaluation of measurement uncertainty” (JCGM, 2012: 2.28; 2.29). The VIM itself provides some examples of type B evaluations, “based on information (i) associated with authoritative published quantity values, (ii) associated with the quantity value of a certified reference material, (iii) obtained from a calibration certificate, (iv) about drift, (v) obtained from the accuracy class of a verified measuring instrument, (vi) obtained from limits deduced through personal experience” (2.29, Examples).
- 12.
The term “standard uncertainty” was introduced by the GUM as “uncertainty of the result of a measurement expressed as a standard deviation” (JCGM, 2008a, 2008b: 2.3.1), where the adjective “standard” here plausibly refers to the choice of formalizing all components of measurement uncertainty with the same mathematical tool, i.e., as standard deviations. Whether this is always a sensible position is an open issue, and in any case for less-than-interval properties other tools need to be adopted, for example the interquartile range for ordinal properties and the entropy for nominal properties (Mari et al., 2020). A basic justification of the choice of standard deviations is implicitly given by the GUM itself, which defines <uncertainty (of measurement)> as “parameter […] that characterizes the dispersion of the values that could reasonably be attributed to the measurand” (2.2.3), thus assuming that, at least in the case of measurement, uncertainty and dispersion can be superposed. That is generally not the case is clear, as this quote shows. “Entropy measures the uncertainty associated with a probability distribution over outcomes. It therefore also measures surprise. Entropy differs from variance, which measures the dispersion of a set or distribution of numerical values. Uncertainty correlates with dispersion, but the two differ. Distributions with high uncertainty have nontrivial probabilities over many outcomes. Those outcomes need not have numerical values. Distributions with high dispersion take on extreme numerical values. The distinction can be seen in stark relief by comparing a distribution that has maximal entropy with one that has maximal variance. Given outcomes that take values 1–8, the distribution that maximizes entropy places equal weight on each outcome. The distribution that maximizes variance takes value 1 with probability 1/2 and value 8 with probability 1/2” (Page, 2018: p. 139, emphasis added). The term “measurement uncertainty” is then taken by the GUM as idiomatic.
- 13.
This classification is less analytical but possibly more conceptually sound than the list of the “many possible sources of uncertainty in a measurement” proposed in the GUM: “(a) incomplete definition of the measurand; (b) imperfect realization of the definition of the measurand; (c) non-representative sampling—the sample measured may not represent the defined measurand; (d) inadequate knowledge of the effects of environmental conditions on the measurement or imperfect measurement of environmental conditions; (e) personal bias in reading analogue instruments; (f) finite instrument resolution or discrimination threshold; (g) inexact values of measurement standards and reference materials; (h) inexact values of constants and other parameters obtained from external sources and used in the data-reduction algorithm; (i) approximations and assumptions incorporated in the measurement method and procedure; (j) variations in repeated observations of the measurand under apparently identical conditions” (JCGM, 2008a, 2008b: 3.3.2).
- 14.
There is an unresolved ambiguity here: Is definitional uncertainty a component of measurement uncertainty, and thus in fact a standard uncertainty, to be combined with the other components, or the lower bound of the result of such a combination? The GUM is quite clear on this matter by considering definitional uncertainty (the GUM calls it “intrinsic”) to be “the minimum uncertainty with which a measurand can be determined, and every measurement that achieves such an uncertainty may be viewed as the best possible measurement of the measurand. To obtain a value of the quantity in question having a smaller uncertainty requires that the measurand be more completely defined” (JCGM, 2008a, 2008b: D.3.4).
- 15.
Measurement uncertainty is dependent on the quality of measurement results given the available information, not in any “absolute” sense. As remarked by Ignazio Lira, “at first sight this is intuitively correct: if two results of the same quantity are available, the one having a smaller uncertainty will be better than the other. However, by itself the uncertainty says nothing about the care put into modelling the measurand, performing the actual measurements and processing the information thus obtained. For example, a small uncertainty may be due to the fact that some important systematic effect was overlooked. Hence, the quality of a measurement can be judged on the basis of its stated uncertainty solely if one is sure that every effort has been taken to evaluate it correctly” (2002: p. 44).
- 16.
As an example, let us consider the task to determine the character written in a given ink pattern, called “optical character recognition” (OCR) in the context of information technology. If the recognition of a given character from a given pattern is not certain, more than one character could be attributed to the pattern, and in the most general case the result of the recognition is a probability distribution over the chosen alphabet (Mari et al., 2020). Hence in this case it is a list of distributions, and not of standard uncertainties, that has to be propagated. Due to the analytical complexity of the problem, the GUM framework includes a numerical procedure for such a propagation of distributions, based on a Monte Carlo method (JCGM, 2008b).
- 17.
For measurands that are quantities, the value is a number that multiplies a unit. In this case the number may actually convey some information about the intended quality of the result through its number of significant digits, so that, for example, “1.23” can be interpreted as including all numbers in the range (1.2250 …, 1.2349 …). This offers a justification for the admission that “the measurement result may be expressed as a single measured quantity value. In many fields, this is the common way of expressing a measurement result” (JCGM, 2012: 2.9, Note 2). Of course, this is less informative than the standard deviation format, except if it is also assumed that the distribution in the range is of a particular kind, such as a uniform distribution.
- 18.
In the context of metrology it is usual to use the expression “measurand definition” (from which, e.g., “definitional uncertainty”, JCGM, 2012: 2.27). Under the assumption that properties of objects are empirical, strictly speaking what can be defined is not a measurand but the concept of it (consider the parallel case of objects: what can be defined is not a rod, but the concept of a rod): a measurand can be instead identified, through a sufficiently specific definition or, more simply but less usefully, a direct reference (“the measurand is the length of that rod” uttered while indicating a rod).
- 19.
The case of temperature is again exemplary of the problems that can be encountered. In the words of Hasok Chang (2007: p. 4): “How do we know that our thermometers tell us the temperature correctly, especially when they disagree with each other? How can we test whether the fluid in our thermometer expands regularly with increasing temperature, without a circular reliance on the temperature readings provided by the thermometer itself? How did people without thermometers learn that water boiled or ice melted always at the same temperature, so that those phenomena could be used as ‘fixed points’ for calibrating thermometers? In the extremes of hot and cold where all known thermometers broke down materially, how were new standards of temperature established and verified? And were there any reliable theories to support the thermometric practices, and if so, how was it possible to test those theories empirically, in the absence of thermometry that was already well established?”.
- 20.
More broadly, the importance of this contribution is also highlighted by the consideration that “Euclid’s Elements, written about 300 BC, has probably been the most influential work in the history of science” (Suppes, 2002: p. 10).
- 21.
Consider, as a significant case, what Michell and Ernst wrote on this matter: “there are two sides to measurement theory: one side (emphasized in the modern era) at the interface with experimental science, the other side (emphasized in the classical) at the interface with quantitative theory” (1996: p. 236). But these are two sides of measure, not measurement. This sentence is excerpted from the introduction that Michell wrote to the English translation of the 1901 paper by Hölder on the axioms of quantity. This confusion was worsened by their choice of translating in the title of Hölder’s paper the German noun “mass” as “measurement” rather than “measure”: they used “The axioms of quantity and the theory of measurement”, instead of “The axioms of quantity and the theory of measure” (Mari et al., 2017).
- 22.
In reference to the black box model we have just discussed, the noun “measure” is sometimes used to refer to each of the three elements of the model: the input property, the process, and the output value. Just as an example, Isaac Newton famously began his Principia with the following two definitions: “Definition I. The Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly. Definition II. The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter conjunctly” (1724: p. 1). Here the concepts <quantity> and <measure> appear to be equivalent. Very interesting on this matter is the following quote from Leonhard Euler: “Whatever is capable of increase or diminution is called magnitude, or quantity. […] Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity. […] Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation. […] So that the determination, or the measure of magnitude of all kinds, is reduced to this: fix at pleasure upon any one known magnitude of the same species with that which is to be determined, and consider it as the measure or unit; then, determine the proportion of the proposed magnitude to this known measure. This proportion is always expressed by numbers; so that a number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit” (1765: pp. 1–2).
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Mari, L., Wilson, M., Maul, A. (2021). Technical and cultural contexts for measurement systems. In: Measurement across the Sciences. Springer Series in Measurement Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-65558-7_3
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