A parallel view of the history of psychophysics

and b, for two weights corresponding to stimuli SA and SB. The average of these two values (a + b)l2 Fechner referred to as die schwelle, or the threshold. Fechner postulated that when the lighter of the two weights SA was judged to be the heavier of the two, it was because the true heaviness of the lighter weight had added to it enough error to make its apparent heaviness greater than the average heaviness of the two weights. Because the lighter weight was perceived to be heavier than the average, it was judged, incorrectly, to be the heavier. By hypothesis, when stimulus SA was lifted the sensory system added a Gaussian error having mean value zero to the true heaviness, a. When an error occurred, the heaviness of SA necessarily exceeded the rejrent located at (a + b)l2. Because (a + b)l2 also equals a + (o a)/2, Fechner reasoned that the amount of error was larger than (b — a)/2. Therefore, the probability of an error of judgment must equal the probability of errors larger than (b — a)l2. The shaded area under the Gaussian probability distribution to the right of the midpoint, (a + b)/2, represents the probability that SA has sufficient error added to its true heaviness to exceed the average heaviness of the two weights and produce an error of judgment. When the larger weight, SB, was judged to be the lighter of the two, the heaviness was judged to be smaller than (o + b)l2, which equals b — (b — o)/2. The total heaviness was less than b by an amount at least as big as (b — d)/2. Thus the error was of the same magnitude for SB as for SA. Due to the symmetry of the Gaussian distribution and the use of the midpoint as a threshold for judgment, the probability of an error in judging SB equals the probability of an error in judging SA. To estimate the probability of an error, Fechner determined the relative frequency of an error by computing the ratio of the observed number of such errors divided by the total number of judgments where such errors could occur. Therefore, an error larger than (b — a)l2 occurred with a probability that was determined through mental judgments self-reports about the status of a sensation. Fechner suggested that by repeated stimulus presentations enough judgments could be gathered to provide a statistically reliable estimate of this unknown probability. For this reason the probability of an error in judgment became an important element of the theory of measurement. For Fechner, estimating the probability of an error of judgment was only a step along the path toward mental measurement. His major concern was to measure the sensitivity of the sensory system in much the same way as determining the sensitivity for any physical measuring device. In the midnineteenth century the statistical quantity called the probable error served this purpose. The probable error equals the distance from the mean of a Gaussian probability distribution to a point that yields an area under the Gaussian distribution equal to 0.25. Plus or minus one probable error from the mean defines the interquartile range containing 50% of the Gaussian probability distribution. Beyond one probable error above the mean is 25% of the distribution which corresponds to large positive deviations from the mean. Another 25% of the distribution consists of large negative deviations that exceed one probable error below the mean. Suppose two weights of 300 g and 318 g are correctly discriminated with probability 0.75. The mean heavinesses of these stimuli are shown as points a and b on the abscissa of Figure 1. In terms of decision making, the average heaviness is a decision criterion that lies midway between the heavinesses corresponding to the two stimuli. The average heaviness corresponds to an interpolated hypothetical stimulus of 309 g. The probable error equals 9 g because 25% of the probability distribution of heavinesses exists between 300 g and 309 g. To measure the sensitivity of the sensory system, Fechner used the reciprocal of the probable error. As the probable error increased, and larger errors became more likely to be added to the true heaviness, this reciprocal decreased, indicating a reduction in sensitivity. For a small probable error, few large deviations from the true heaviness occurred and the measure of sensitivity increased. The measurement of the probable error, based on Herbart's and Weber's idea of a threshold, allowed psychological scientists to determine whether one judge could discriminate a smaller difference between two stimuli than another, or whether, for the same judge at different times, the size of the physical difference needed for 75% correct discrimination varied. In this way Fechner showed how to investigate individual differences and also how to examine variations within an individual to such causes as fatigue, time of day, and order of stimulus presentation. The major point of Fechner's psychophysics is that the wholly invisible error of the sensory measuring device was made measurable through an extension of Gauss's Theory of Error. The theory was later recreated by engineers who were entirely unaware of Fechner's contribution to the founding of psychology. This new rendition is a form of signal detection theory known as ideal observer theory. Somehow this quite stunning idea became lost to psychology. Many students trained by Wundt in Leipzig took up residence in the United States. Yet their textbooks did not present Fechner's theory. Perhaps the loss of Fechner's theory is a consequence of the loss of many young European students in two World Wars, compounded by the zealous grip on experimental psychology of the antiquantitative, behavioristic forces of the twentieth century. The antiquantitative bias is revealed in a telling comment found in the introduction to Experimental psychology: A manual of laboratory practice by E. B. Titchener (1901), scion of Wundt's line in American experimental psychology: "Knowledge of elementary mathematics is a part of a man's general scientific outfit: but one may work a lifetime, and with success, in psychology, without needing the knowledge" (p. xxiii). Fechner's great insight did not entirely disappear, but the band of psychological scientists dedicated to the measurement of psychological phenomena grew smaller as the statistical/mathematical models of additive effects came to dominate many psychologists' methodology. The story of the theorists and experimentalists who continued beyond Fechner's pioneering work is told in Part I of Link (1992, "Psychophysical origins of difference and similarity"). Part II presents the theory that generates feeling from sensation and resolves many previous psychophysical paradoxes by showing how Weber's constant, Fechner's Law, and Stevens's exponents are derived and related.