Weber–Fechner law
The Weber -Fechner law states that the perceived intensity of sensations is proportional to the logarithm of the objective intensity of the physical stimulus.
1834 saw the physiologist Ernst Heinrich Weber, that a sense organ from a certain intensity value a change is registered (differential visibility threshold, English: just noticeable difference = just noticeable difference ), as difference? R to the previous stimulus R in a certain, constant ratio k is this:
When you touch the required relative difference R / R is, in Weber's experiments about 3 percent of skin pressure, the brightness seeing about 1 to 2 percent of the light intensity. When you taste the concentration must rise by 10 to 20 percent in order to be perceived as stronger.
For example, do you recognize a relative weight difference of about 2 percent of an investment in the resting hand object. It is the increase in weight of an object from an initial 50 g ( grams) only true if the weight is increased by 1 g to 51 g. According to 5000 g weight must increase by 100 g to look heavier.
The physicist and founder of psychophysics Gustav Theodor Fechner extended Weber's law in 1860 formally by integration under the assumption that k is constant and independent of R:
R0 is a constant of integration, which usually sets the threshold stimulus. (2 ) states that in an exponential increase in stimulus intensity their sensation in the sensory organ grows only linearly. Here c is the dependent on the particular type of stimulus size.
Due to the logarithmic adaptation to the human eye can sense impressions of brightness between dusk and bright sunshine of up to 10.5 orders of magnitude (2 × 10-5 ... ~ 106 cd / m²) luminance of physical bridge.
The magnitude (mag ) is a brightness size, the difference between each brightness level (size class) is about 2,512 -fold. A freiäugig barely visible 6th mag star ( 6 mag ) is opposite the Sun (-25 mag) weaker by 31 size classes or 12 ¼ orders of magnitude. An experienced astronomer can in the visual photometry differences in brightness of two stars of only a few percent perceive ( Argelander step estimation method).
The perceived pitch of a musical tone depends logarithmically on the fundamental frequency. A doubling of the fundamental frequency causes the change of the pitch by one octave. In contrast, the Weber- Fechner law does not apply to the perceived volume, the so-called loudness. This follows for medium and high sound pressures the Stevens 's power law: An increase in the sound pressure by a factor (10 dB ) causes a doubling of loudness.
In the temperature sense, however, the reaction of the thermal receptors takes approximately linearly with stimulus size. For here is less the " measurement" of the temperature important, but rather a warning against burning or freezing to death before. The same applies to the perception of pain.
In order to illustrate
This graphic illustrates the light sensitivity of the human eye. On the left is for the same AE as different intensities of light sensation, right is with the same AH the different energy of the light. This logarithmic relationship describing the Weber- Fechner law.
Power law
Stanley Smith Stevens introduced in 1957 found that the extension of Weber 's law ( 1) to equation (2 ) is too general. Considering the dependence of the reaction force E from the size of the stimulus, it follows from ( 1):
The integration of this relationship leads to Stevens 's power function:
The constant c is derived from the integration results from the two constants of integration. For k < 1, it is similar to the logarithmic Weber- Fechner law. For the perception of brightness is k ≈ 0.33.
Microeconomics
In microeconomics, the phenomenon is found again as perceptibility. Of interest, it is in the investigation of indifference curves in their characteristics continuity and transitivity. The microeconomic theory assumes that indifference curves are continuous, decreasing and convex curved.
The practical problem that a person marginal differences in color or temperature, etc. can not perceive means that one must formulate the Transitivitätsannahme little looser.