The function describing the relationship between values on the stimulus continuum and responses would then be of the form SUppose further that the numerical judgment, J, is some function f of the psychological value. If f is also a power function, it follows that J=a4J m• (3)

In previous studies, judgments of ratios and differences in subjective magnitude have yielded similar orders, consistent with a hypothesis that a single perceived relation underlies both judgment tasks. In the present research, 15 subjects estimated heaviness differences between 28 pairs of eight weights and each of 8 groups of 10 subjects evaluated heaviness ratios of eight variable stimuli with respect to a different standard stimulus. Presenting stimuli that were equally spaced on a cube-root scale of weight enhanced expected ordinal discrepancies between ratio and difference estimates, and employing independent groups for each standard stimulus in ratio estimation eliminated a possible bias due to varying standards within the presentation sequence. Differences in orders of ratio and difference estimates together with differences in scales obtained from non-metric analyses in terms of a difference model indicated that the judgments were based on two perceived relations that are ordinally consistent with arithmetic operations of ratios and differences. A ratio scale of heaviness was derived from the combined orders of subjective ratios and differences.

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Judged magnitudes of difference in area of paired circles and magnitude estimations of the circles making up the pairs were obtained from 11 5s. The difference judgments were subjected to nonmetric scaling, and a one-dimensional solution was obtained. The relationships between scale values and physical area and between judged difference and derived distance were each characterized by power functions. The product of exponents from the two functions closely predicted judgments of individual circles. Judgments of differences between paired weights were subjected to the same analysis. The relationships between scale values and physical weight and between judged difference and derived distance were power functions for both pooled data and data from individual 5s, indicating that input and output transformations in magnitude estimation are power functions.

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