Thurstonian psychophysics: Case III

Thurstonian psychophysics is re-examined from the viewpoint of a theory of “random comparisons” which generalize the notion of “discriminal differences.” The assumption of normality which appears so crucial to all varieties of Thurstone models is not invoked but becomes a consequence of our theory—at least this is so under independence assumptions similar to those employed in conventional derivations of Case III models. If nothing else, the theoretical scheme developed in this work clarifies the status of Thurstone's assumption of normality, placing it in the deeper probabilistic context of stability. Under reasonable side assumptions we are also able to arrive at satisfactory conclusions as to the uniqueness of the scales (i.e., mean and standard deviation) which appear in Case III of the Law of Comparative Judgement.