Stevens' formula

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This law (from the 1950's) is more founded on experiments where two stimuli of different intensity were given, and the victim asked to quantify how different. For example: "is this sound twice as strong as that sound?"

This kind of question is difficult to answer, and many people find it a bit absurd. But if you coerce an answer out of enough people, there is system in the answers though there is of course a lot of random variation.

Stevens found that such results from different sensory modalities varied too much in "steepness" to be fitted by the Webner-Fechner law. Instead he introduced a formula with one more parameter, and therefore more flexible:

R = k (S-S₀)^a

(the a should be an alpha sign really, but that is difficult on the Web).

If we take the logarithm of both sides of this formula, we get

log R = a log(S-S₀) + log k

ie there is a rectilinear relationship between log(S-S₀) and log R, with the slope of the line determined by a. Data of this type are therefore usually presented in this kind of bilogarithmic diagram:
 

This diagram gives a rough idea of things. Pain has a high value of a, reflected in a steep curve. In other words, once a stimulus is strong enough to elicit pain, the pain rapidly becomes stronger as the stimulus becomes stronger. The other modalities shown have successively lower a values, which means that they can cover much wider ranges of stimulus intensity.

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Once Stevens' formula was established in psychophysics, it also got popular for describing results of neurophysiological experiments relating stimulus intensity to objective response, for example frequency of actions potentials

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