A conceptual analysis of measurement can properly begin by formulating the two fundamental problems of any measurement procedure. The first problem is that of representation, justifying the assignment of numbers to objects or phenomena. We cannot literally take a number in our hands and ’apply’ it to a physical object. What we can show is that the structure of a set of phenomena under certain empirical operations and relations is the same as the structure of some set of numbers under corresponding arithmetical operations and relations. Solution of the representation problem for a theory of measurement does not completely lay bare the structure of the theory, for there is often a formal difference between the kind of assignment of numbers arising from different procedures of measurement. This is the second fundamental problem, determining the scale type of a given procedure.
Counting is an example of an absolute scale. The number of members of a given collection of objects is determined uniquely. In contrast, the measurement of mass or weight is an example of a ratio scale. An empirical procedure for measuring mass does not determine the unit of mass. The measurement of temperature is an example of an interval scale. The empirical procedure of measuring temperature by use of a thermometer determines neither a unit nor an origin. In this sort of measurement the ratio of any two intervals is independent of the unit and zero point of measurement.
Still another type of scale is one which is arbitrary except for order. Moh’s hardness scale, according to which minerals are ranked in regard to hardness as determined by a scratch test, and the Beaufort wind scale, whereby the strength of a wind is classified as calm, light air, light breeze, and so on, are examples of ordinal scales.
A distinction is made between those scales of measurement which are fundamental and those which are derived. A derived scale presupposes and uses the numerical results of at least one other scale. In contrast, a fundamental scale does not depend on others.
Another common distinction is that between extensive and intensive quantities or scales. For extensive quantities like mass or distance an empirical operation of combination can be given which has the structural properties of the numerical operation of addition. Intensive quantities do not have such an operation; typical examples are temperature and cardinal utility.
A widespread complaint about this classical foundation of measurement is that it takes too little account of the analysis of variability in the quantity measured. One important source is systematic variability in the empirical properties of the object being measured. Another source lies not in the object but in the procedures of measurement being used. There are also random errors which can arise from variability in the object, the procedures or the conditions surrounding the observations.
The analytic hierarchy and analytic network measurement processes: Applications to decisions under Risk
