Academic journal article Cartography & Geographic Information Systems

Rethinking Levels of Measurement for Cartography

Academic journal article Cartography & Geographic Information Systems

Rethinking Levels of Measurement for Cartography

Article excerpt

ABSTRACT: Stevens' measurement levels (nominal, ordinal, interval and ratio) have become a familiar part of cartography and GIS. These levels have been accepted unquestioned from publications in social sciences dating from the 1940s and 1950s. The Stevens taxonomy has been used to prescribe appropriate symbolism (or analytical treatment) to each scale of measurement. This paper reviews the process by which these levels became a part of cartography, as well as subsequent literature that cartographers have all but ignored over the intervening four decades. The paper concludes that the four levels of measurement are not adequate to cover the circumstances of cartography, and that attribute issues alone do not provide a sufficient guide to symbolism or analytical treatment. A broader framework for measurement must be considered, including the relationships of control that constrain variation in one component to permit measurement of another. An informed use of tools does not depend on numbers alone, but on the whole "measurement framework," the system of objects, relationships and axioms implied by a given system of representation.

How Measurement Levels Reached Cartography

The approach to measurement in certain social sciences (including geography) is still strongly influenced by Stevens' (1946) paper in Science. Stevens' levels of measurement form an important foundation for textbooks in spatial analysis (Unwin 1981) and in cartography (Robinson et al. 1995; Muehrcke and Muehrcke 1978; Dent 1993). While there has been continued development of the theory of measurement (Khurshid and Sahai 1993), the debate on measurement in cartography and GIS literature remains stuck in 1946. This phenomenon is interesting because it is followed by an absorption of a new set of ideas, and resistance to further changes. It also is important because the measurement concepts serve as foundations for instruction. This first section will review the process leading up to Stevens' paper and how it diffused into cartographic practice.

History of Theories of Measurement

The "classical" school of measurement has its origins in Aristotle's metaphysics and continues through Newton to include the physical sciences at the end of the nineteenth century. In the classical view, measurement discovered the numerical ratio between a standard object and the one measured.

Clifford, a mathematician in the nineteenth century, used a definition of measurement not much different from Euclid's: "every quantity is measured by the ratio which it bears to some fixed quantity, called the unit" (Clifford 1870). The property was seen to be "inherent" in the object, and the relationship took on all the mathematical properties of numbers without question (Michell 1993).

Let us take the attribute "length." Every entity in space can be measured by comparing its length to some other length. If we adopt a "standard" measuring rod, the ratio between the length of the rod and the objects measured embodies a number implicit in the relationship. The realm of numbers is then, in turn, seen to be derived from these basic relationships of ratios.

Through a series of theoretical developments, this classical approach was overthrown and completely replaced by a "representational" school in the first half of the twentieth century (Michell 1993). The representationalists impose a firm separation between the empirical world and numbers, as first expressed by Bertrand Russell (Russell 1897; 1903). In the place of ratios and rational numbers as the core of mathematics, set theory permitted a more generic range of relationships. According to Russell:

   Measurement of magnitudes is, in its most general sense, any method by
   which a unique and reciprocal correspondence is established between all or
   some of the magnitudes of a kind and all or some of the numbers, integral,
   rational, or real, as the case may be . … 
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