A formalization of measurement scale forms

The general measurement scale hierarchy developed by S. S. Stevens has had widespread use and discussion, but has suffered from various criticisms, many of which may be traced to the informal descriptions and minimal mathematical formulations used in the original definitions and claims. In this paper, a mathematical model of measurement and specific scale forms is developed, using the minimum mathematical structure necessary to define each scale type. Measurement is defined, and a series of restrictions are applied to measurements to yield a precisely specified hierarchy of measurement scale forms including nominal, ordinal, interval, and ratio scales, as well as several others. Equivalences are mathematically denned for each scale form, and theorems are presented specifying the sets of transformations under which equivalence is preserved, thus yielding the scale form invariances discussed by Stevens. The hierarchy of measurement scale forms is summarized in a pictorial ‘spectrum’ diagram.

The development of the model in terms of the structure of possible measurement comparisons (e.g., whether one can establish that one item is ‘less’ than another, or, alternatively, establish how much the one is less than the other, etc.) is in marked contrast to Stevens’ definition in terms of scale invariances. This approach avoids several criticisms raised against Stevens’ approach. One might hope that the alternative approaches would yield corresponding results, and indeed this is found to be generally true. Those cases where differences do occur, however, prove to be vital, and yield several striking unanswered questions about scale forms and their invariances.

Notes

Present address: Department of Electrical Engineering, Purdue University, Lafayette, Indiana.