The Power Law and Subjective Scales of Number

Ekrnan (1964) has offered a derivation of the psychophysical power law from Fechner's conjecture of a logarithmic relationship of subjective magnitude to stimulus intensity. The derivation assumes that Fechner's "law" describes subjective responses to numbers as well as reactions to conventional prothetic domains. Thus, argues Ekman, magnitude estimation requires S to match subjective impressions of the stimuli under test to his internal scale of number. The equating of rwo logarithmic functions produces mathematically a power relationship between their arguments. Some years ago, Dr. Ethel Matin, Dr. Truett Allison, and I became interested in possible subjective scales for numbers. W e posed the problem raised by Ekman in a more general form and asked whether subjective responses to number distort magnitude or caregory scales in any fashion. To answer the question, we had people scale numbers by magnitude estimation and by category rating. The instructions for magnitude estimations ran along the following lines: This is an experiment to test your reactions to numbers. Whenever I say the number "37," you must respond with "83." If I give you a number that seems 8 times larger than 37, you must respond with 664, which is 8 times larger than 83. If I say a number char seems about half as large as 37, you must say 42. In other words, you should reply with a number which bears the same proportion to 8 3 as the number I say bears to 37. Now I will read numbers to you very quickly. Answer just as fasr as you can. You won't have time to calculate anything so do your best and don't worry about being consistent. One E then read a random, replicated sequence of numbers at a rare of about 1 every 2 sec. Another E wrote down S's replies. Then, after receiving appropriate instructions, S quickly rated numbers on a 7-point category scale. At the end of the experiment, each S felt that the task was fairly easy. The most harassed participant in the srudy was the E who recorded the responses. Neither the magnitude estimates nor the category ratings gave any indication of nonlinear subjective responses to number. Ekman's theory could account for the linear magnitude estimates. If number were a prothetic continuum, however, the category ratings should have been non-linear and they were not. The simplest interpretation of the results is that Ss have a nicely linear subjective scale of number. Acceptance of this interpretation implies rejection of Ekman's derivation of the power law from Fechner's conjecture.