Psychophysical Laws:
Fechner's vs. Stevens'

Background:

Psychophysics is the study of the relation between real world stimuli and the perception of those stimuli. One of the main goals of psychophysics is to determine mathematical formulae which allow you to predict the perceptual response to a physical object. For example, psychophysics would allow you to predict how bright a light of a given physical intensity would appear to be. Two common psychophysical laws are Fechner’s Law and Stevens’ Power Law.

In this study you will judge how bright several color patches are. According to Fechner’s law, the perceived (psychological) brightness of the color patches should be a logarithmic function of the physical intensity of the color patches:

ψ = k ln(I / I₀)

Where ψ is the psychological brightness of the color patch, I is the physical intensity of the color patch (as measured by a light meter), I₀ is the physical intensity of a color patch that is just noticeable, and k is a constant (Weber’s constant.) Fechner’s law is based on the assumption that each just noticeable difference (JND) is psychologically equivalent. Therefore the JNDs can be used as a ruler for determining the psychological intensity of the stimulus.

According to Stevens’ Power law, the perceived brightness of the color patches should be a power function of the physical intensity of the color patches:

ψ = c Iⁿ

Where ψ is the psychological brightness of the color patch, I is the physical intensity of the color patch (as measured by a light meter), n is the power to which the intensity is raised, and c is a constant. Stevens’ power law is not based on an assumption – rather, it is based on the empirical observations of the relation between the psychological brightness and the physical intensity of many different types of stimuli.

The goal of this study is for you to determine which of these two laws, Fechner’s or Stevens’, better describe the relation between psychological brightness and the physical intensity of color patches based on data that you provide.

The Study:

Below you will see three circles on a blue background. The two smaller circles are the anchors for your judgments of the brightness of the larger circle. The smaller black circle is arbitrarily set to a brightness of one, while the smaller white circle is arbitrarily set to a brightness of 100. Relative to the two smaller circles, you are to enter an integer (whole) number between 1 and 100 inclusive that indicates how bright you think the larger circle is. For example, if you thought the brightness of the larger circle was exactly half way in between the brightness of the black and of the white circle, you should enter 50 (half way between 1 and 100.) If you thought the brightness of the larger circle was three fourths as bright as the smaller white circle, you should enter 75 (three fourths of 100.)

Across the trials, the larger circle will have several different physical intensities. Each of the different physical intensities will be presented four times. Once you have judged the psychological brightness of each color patch (it should not take too long – no more than a couple of minutes to judge the brightness of all of the color patches,) you will see a table of your results. The table will include the approximate physical intensities and each of your four judgments of the brightness of each physical intensity. The table will also include the mean of your four judgments of each physical intensity.

A graph of the approximate physical intensity (X axis) versus the mean of your psychological intensities (Y axis) will be presented. On the graph will be two curves. The red curve is the best fitting (least squares regression curve) logarithm function for your data – this is an indication of how well Fechner’s law describes your data. The blue curve is the best fitting (least squares regression curve) power function for your data – this is an indication of how well Stevens’ Power law describes your data.

To quantify how well each curve describes your data, the coefficient of determination, r², is reported for each curve. r² tells us the proportion of the variability in the Y axis variable that is explained by variability in the X axis variable. In this case, r² tells us how much of the differences in the psychological intensity is explained by differences in the physical intensity. The closer r² is to 1.000, the better the curve describes the data.

Collect Some Data!: