Weber-Fechner Law
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The Weber - Fechner law attempts to describe the human perception of various physical stimuli. Ernst Heinrich Weber (1795-1878), was one of the first people to quantitatively study the human response to a physical stimulus. Gustav Theodor Fechner (1801-1887) later offered an elaborate theoretical interpretation of Weber's findings, which he called simply Weber's law, though his admirers made the law's name a hyphenate.
In most cases, Stevens' power law gives a more accurate description.
Fechner believed that Weber had discovered the fundamental principle of mind/body interaction, a mathematical analog of the function Rene Descartes once assigned to the pineal gland.
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The case of weight
In one of his classic experiments, Weber gradually increased the weight that a blindfolded man was holding and asked him to respond when he first felt the increase. Weber found that the response was proportional to a relative increase in the weight. That is to say, if the weight is 1 kg, an increase of a few grams will not be noticed. Rather, when the mass in increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold is also doubled. This kind of relationship can be described by a differential equation as,
- <math> dp = k\frac{dS}{S} <math>
where dp is the differential change in perception, dS is the differential increase in the stimulus and S is the stimulus at the instant. A constant factor k is to be determined experimentally.
Integrating the above equation
- <math> p = k \ln{S} + C <math>
with C is the constant of integration, ln is the natural logarithm.
To determine C, put p = 0, i.e. no perception; then
- <math> C = -k\ln{S_0} <math>
where <math>S_0<math> is that threshold of stimulus below which it is not perceived at all.
Therefore, our equation becomes
- <math> p = k \ln{\frac{S}{S_0}}. <math>
The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if the perception is altered in an arithmetic progression (i.e. add constant amounts) the corresponding stimulus varies as a geometric progression (i.e. multiply by a fixed factor).
The point is that this logarithmic relationship is valid, not just for the sensation of weight, but for other stimuli as well.
The case of vision
The eye senses brightness logarithmically. Hence stellar magnitude is measured on a logarithmic scale. This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits. An increase in 5 magnitudes corresponds to a decrease in brightness by a factor 100.
The case of sound
Still another logarithmic scale is the decibel scale of sound intensity (see the discussion page for details). And yet another is pitch. In the case of perception of pitch, humans hear pitch in a logarithmic or "geometric" ratio-based fashion. For instance, the "pitch distance" between 100 Hz and 150 Hz sounds the same as 1000 Hz and 1500 Hz. The frequency of corresponding notes of adjacent octaves differ by a factor of 2. For notes spaced equally apart to the human ear, the frequencies are related by a multiplicative factor. For instance, for a 12-tone scale in equal temperament, this factor is <math> \sqrt[12]{2} <math> (twelfth root of 2). So the frequency of the A# note is the frequency of the A times the 12th root of 2, for any octave.
Economics
In 1889, the Austrian economist Friedrich Wieser, in "Natural Value," coined the phrase marginal utility for a closely-related phenomenon -- the satiation of human appetite for identical increments of a good.
"Any one who has just taken a certain quantity of food of a certain kind will not immediately have the same strength of desire for another similar quantity," he wrote. "Within any single period of want every additional act of satisfaction will be estimated less highly than a preceding one obtained from a quantity of goods equal in kind and amount."
A non-Fechnerian interpretation of Weber's results
In 1890, the American psychologist William James described Fechner's writings on the subject of Weber's results as "patient whimsies" and said it would be a pity if Fechner should "compel all future students" of psychology "to plough through the difficulties, not only of his own works, but of the still drier ones written in his refutation."
James saw Weber's law as an accurate generalization as to the friction in the neural machinery.
"If our feelings [of weight, sight, sound, etc.] resulted from a condition of the nerve molecules which it grew ever more difficult for the stimulus to increase, our feelings would naturally grow at a slower rate than the stimulus itself. An ever larger part of the latter's work would go to overcoming the resistances, and an ever smaller part to the realization of the feeling-bringing state."