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Sometimes as a stimulus gets stronger, we become increasingly less sensitive to the stimulus changes. Think of the results from the Dot Brightness or Tone Loudness magnitude estimation experiments. Sometimes as the stimulus intensity increases, our sensitivity keeps roughly constant. Think of the judging line lengths magnitude estimation results. Sometimes as the stimulus intensity increases, our sensitivity increases as well. Think of painful stimuli. Oddly, I did not develop such an experiment.

Stevens (1957, 1961) developed an equation to try to encapsulate this full range of possible data sets. It is called Stevens’ Power Law and it is as follows:

P = cIb

In this equation, P is equal to the perceived magnitude—that is, how bright we perceive a light to be our how sweet we perceive a sugar solution to be. I is equal to the intensity of the actual stimuli. Thus, at the simplest level, our perception is a function of the physical intensity of the stimulus. However, there are two other parts of the equation that help explain the relation between perception and the physical stimulus. The letter c equals a constant, which will be different for each sensory modality. The constant also allows you to scale your measure appropriately. For example, both the Fahrenheit and Celsius temperature scale measure the same underlying property but do so with a different scale.

The exponent b equals the power to which the intensity is raised. It is this exponent b which allows for response compression, linear response, or response expansion. Response compression occurs when b is less than 1, like the results from the dot brightness or tone loudness experiments. Linear responses occur when b = 1, like the line length experiment. Response expansion occurs if b is greater than one, like for painful stimuli. Thus, Steven’s Power Law equation can account for all of these types of subjective responses.


Full Screen Mode

To see the illustration in full screen, which is recommended, press the Full Screen button, which appears at the top of the page.

Illustration Tab

On the Illustration tab, you can adjust the parameters of the Power Law on up to three different lines to see how the power law can explain response compression, linear responses, and response expansion.


Below is a list of the ways that you can alter the model. The settings include the following:

You can display up to three lines on this illustration. Each line can be handled in the same way.
Line #: check to display. Uncheck not to display. The # refers to the particular line.
C #: the c value for the power law for the # line. Adjust to see how this parameter alters the power law output.
B #: the b exponent for the power law. Adjust to see how changing the exponent changes the shape of the relationship between intensity (I) and perceived magnitude (P).


Pressing this button restores the settings to their default values.