**2**

What Is a Ratio in Ratio Scaling?

What Is a Ratio in Ratio Scaling?

R. Duncan Luce **Irvine Research Unit in Mathematical Behavioral Science University of California, Irvine**

**THE PROBLEM**

**Ratio Scales and Ratio Scaling**

The terms *ratio scale*, apparently first introduced by Stevens ( 1946, 1951),
and *ratio scaling*, apparently first introduced by Krantz ( 1972) but closely
related to previous phrasing, denote different but interrelated things. Although Krantz was very clear on the matter,^{1} the similarity of the terms
seems to invite confusion and confounding. My goal here is to explicate
some aspects of the differences and relations.

A ratio scale concerns one aspect of numerical measurement representations^{2} for a certain class of one-dimensional, empirical structures.
In particular, it refers to those cases where the numerical representation
of a qualitative structure of stimuli is uniquely specified up to multiplication
by a positive constant. The most familiar physical examples of ratio scales
are length, mass, and time intervals. They are all examples of what are
called *extensive structures*, and they have in common two primitives. The
first is a binary ordering relation ≳ that reflects the ordering induced on

^{1}

*ratio scaling*' to refer to a family of interrelated psychophysical methods, discussed and classified by Stevens [ 1975]. The quotation marks are used to emphasize the distinction between the term '

*ratio scaling*' and the term

*ratio scale*, the latter having its usual technical meaning in measurement theory . . . . 'Ratio scaling' need not lead to ratio scales."

^{2}

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