The Irony of Measurement by Subjective Estimations

In 1948 S. S. Stevens, in his famous Science article, proposed a theory of measurement that radically differed from the dominate theory of the time. The dominate theory held that all strong forms of scientific measurement— for example, those that yielded ratio scales—had to be based on an observable ordering and an observable commutative and associative operation. Stevens proposed different criteria and introduced his method of magnitude estimation. Stevens as well as measurement theorists considered his method to be radically different from those based on commutative and associative operations. Although his method was controversial, it became a standard tool in the behavioral sciences. This article argues that Stevens' method, together with implicit assumptions he made about the scales of measurement it generated, is from a mathematical perspective the same as the measurement process based on commutative and associative operations. The article also provides a theory of qualitative numbers and shows an interesting relationship between qualitative numbers and Stevens' method.