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| In [[psychophysics]], the '''Weber–Fechner law''' combines two different laws of human perception. [[Ernst Heinrich Weber]] (1795–1878) was one of the first people to approach the study of the human response to a [[Stimulus (physiology)|physical stimulus]] in a [[Quantitative research|quantitative]] fashion.<ref>Ross, H.E. and Murray, D. J.(1996)(Ed. and Transl.) ''E.H.Weber on the tactile senses''. 2nd ed. Hove: Erlbaum (UK) Taylor & Francis.</ref> '''Weber's law''' states that the [[just-noticeable difference]] between two stimuli is proportional to the magnitude of the stimuli. [[Gustav Fechner|Gustav Theodor Fechner]] (1801–1887), a scholar of Weber, later used Weber's findings to construct a psychophysical scale in which he described the relationship between the physical magnitude of a [[Stimulus (physiology)|stimulus]] and its (subjectively) perceived intensity. '''Fechner's law''' (better referred to as Fechner's scale) states that subjective sensation is proportional to the logarithm of the stimulus intensity. Fechner scaling has been mathematically formalized. In fact, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to log<sub>10</sub> (actual intensity measured with an accurate nonhuman instrument)
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| Some authors use the term "Weber–Fechner law" to mean Weber's law, and others use it for Fechner's law. The use of the term "Weber–Fechner law" was criticised as a misnomer by [[Ewald Hering]] and should probably be deprecated.
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| == Derivation of Fechner's law for weight perception == | |
| {{Refimprove section|date=February 2011}}
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| Weber found that the [[just noticeable difference]] between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can (only just) be distinguished from that of 100 g, the jnd (or differential threshold) is 5 g, or in the SI system, a force or weight of 0.005Kg N. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight (any weight) seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction" for detecting changes in weight. Other discrimination tasks, such as detecting changes in brightness, or in tone height (pure tone frequency), or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change.
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| This kind of relationship can be described by the [[ordinary differential equation|differential equation]]
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| :<math> dp = k \frac{dS}{S}, \,\!</math>
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| where ''dp'' is the differential change in perception, ''dS'' is the differential increase in the stimulus, and ''S'' is the instantaneous stimulus. The [[parameter]] ''k'' is to be estimated using experimental data. | |
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| Integrating the above equation gives
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| :<math> p = k \ln{S} + C, \,\!</math>
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| where <math>C</math> is the [[constant of integration]] and ''ln'' is the [[natural logarithm]].
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| To solve for <math>C</math>, put <math>p = 0</math>, i.e., no perception; then subtract <math>k\ln{S_0}</math> from both sides and rearrange:
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| :<math> C = -k\ln{S_0}, \,\!</math>
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| where <math>S_0</math> is that threshold of stimulus below which it is not perceived at all.
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| Substituting this value in for <math>C</math> above and rearranging, our equation becomes:
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| :<math> p = k \ln{\frac{S}{S_0}}. \,\!</math>
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| The relationship between stimulus and perception is [[logarithmic scale|logarithmic]]. This logarithmic relationship means that if a stimulus varies as a [[geometric progression]] (i.e., multiplied by a fixed factor), the corresponding perception is altered in an [[arithmetic progression]] (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e., 3 x 1), the corresponding perception may be two times as strong as its original value (i.e., 1 + 1). If the stimulus is again tripled in strength (i.e., 3 x 3 x 1), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1). Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law (see [http://cogprints.org/4094/ link1] or [http://www.bio-balance.com/Ref.htm link2]).
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| Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all jnds are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.<ref>Heidelberger, M. (2004)''[http://books.google.com/books?hl=en&lr=&id=t8PW5CSFADAC Nature from within: Gustav Theodor Fechner and his psychophysical worldview]''. Transl. C. Klohr. Pittsburgh, USA: University of Pittsburg Press.</ref><ref>{{cite journal | last1 = Masin | first1 = S.C. | last2 = Zudini | first2 = V. | last3 = Antonelli | first3 = M. | year = 2009 | title = Early alternative derivations of Fechner's law | url = http://www.psy.unipd.it/~masin/Masin-2009c.pdf| journal = J. History of the Behavioral Sciences | volume = 45 | issue = | pages = 56–65 | doi = 10.1002/jhbs.20349 }}</ref> Most researchers nowadays accept that a power law is a more realistic relationship, or that a logarithmic function is just one of a family of possible functions.<ref>{{cite journal | last1 = Murray | first1 = D.J. | year = 1993 | title = A perspective for viewing the history of psychophysics | url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6534920 | journal = Behavioral and Brain Sciences | volume = 16 | issue = | pages = 115–186 }}</ref>
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| Other sense modalities provide only mixed support for either Weber's law or Fechner's law.
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| == The case of sound ==
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| Weber's law does not quite hold for [[loudness]]. It is a fair approximation for higher intensities, but not for lower amplitudes.{{Citation needed|date=December 2012}}
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| === "Near miss" of Weber's law in the auditory system ===
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| Weber's law does not hold at perception of higher intensities. Intensity discrimination improves at higher intensities. The first demonstration of the phenomena were presented by Riesz in 1928, in Physical Review. This deviation of the Weber's law is known as the "near miss" of the Weber's law. This term was coined by McGill and Goldberg in their paper of 1968 in Perception & Psychophysics. Their study consisted of intensity discrimination in pure tones. Further studies have shown that the near miss is observed in noise stimuli as well. Jesteadt et al. (1977)<ref>Jesteadt, Walt, Craig C. Wier, and David M. Green. "Intensity discrimination as a function of frequency and sensation level." The Journal of the acoustical society of America 61 (1977): 169.</ref> demonstrated that the near miss holds across all the frequencies, and that the intensity discrimination is not a function of frequency, and that the change in discrimination with level can be represented by a single function across all frequencies.
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| == The case of vision ==
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| The eye senses [[brightness]] approximately logarithmically over a moderate range (but more like a [[power law]] over a wider range), and [[stellar magnitude]] is measured on a logarithmic scale.<ref name=bhatia>
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| {{cite book
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| | author = V. B. Bhatia
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| | title = Astronomy and astrophysics with elements of cosmology
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| | publisher = CRC Press
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| | year = 2001
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| | isbn = 978-0-8493-1013-3
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| | page = 20
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| | url = http://books.google.com/books?id=k4XRQpKV9kgC&pg=PA20
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| }}</ref>
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| 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 of 100.<ref name=bhatia/>
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| Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.<ref>
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| {{cite journal
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| |doi=10.1007/s00245-005-0850-1
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| |author1=Jianhong (Jackie) Shen
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| |author2=Yoon-Mo Jung
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| |title=Weberized Mumford–Shah model with Bose–Einstein photon noise
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| |journal=Appl. Math. Optim.
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| |volume=53
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| |issue=3
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| |pages=331–358
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| |year=2006
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| |url=http://www.springerlink.com/content/172467181w543245/?p=cab484b467b64e729a4666703e273f95&pi=3}}
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| </ref><ref>
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| {{cite journal
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| |author=Jianhong (Jackie) Shen
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| |title=On the foundations of vision modeling I. Weber's law and Weberized TV (total variation) restoration
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| |journal=Physica D: Nonlinear Phenomena
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| |volume=175
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| |issue=3/4
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| |pages=241–251
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| |year=2003
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| |doi=10.1016/S0167-2789(02)00734-0
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| }}</ref>
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| === "Near miss" of Weber's law in visual regularity perception ===
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| Perception of Glass patterns and mirror symmetries in the presence of noise follows Weber's law in the middle range of regularity-to-noise ratios (''S''), but in both outer ranges, sensitivity to variations is disproportionally lower. As Maloney, Mitchison, & Barlow (1987)<ref>Maloney, R. K., Mitchison, G. J., & Barlow, H. B. (1987). Limit to the detection of Glass patterns in the presence of noise. ''Journal of the Optical Society of America A, 4,'' 2336-2341.</ref> showed for Glass patterns, and as van der Helm (2010)<ref>van der Helm, P. A. (2010). Weber-Fechner behaviour in symmetry perception? ''Attention, Perception, & Psychophysics, 72,'' 1854-1864.</ref> showed for mirror symmetries, perception of these visual regularities in the whole range of regularity-to-noise ratios follows the law ''p'' = ''g''/(2+1/''S'') with parameter ''g'' to be estimated using experimental data.
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| == The case of numerical cognition ==
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| Psychological studies show that numbers are thought of as existing along a mental number line.<ref>
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| {{cite journal |doi=10.1038/2151519a0 |author=Moyer R.S., Landauer T.K. |title=Time required for judgements of numerical inequality |journal=Nature |volume=215 |issue=5109 |pages=1519–20 |date=September 1967 |pmid=6052760 }}
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| </ref> It becomes increasingly difficult to discriminate among two places on a number line as the distance between the two places decreases—known as the ''distance effect''.<ref>
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| {{cite journal |author=Longo M.R., Lourenco S.F. |title=Spatial attention and the mental number line: evidence for characteristic biases and compression |journal=Neuropsychologia |volume=45 |pages=1400–6 |year=2007 |doi=10.1016/j.neuropsychologia.2006.11.002 |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T0D-4MHPC4G-1&_user=201547&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000014058&_version=1&_urlVersion=0&_userid=201547&md5=ed6a3e2bc764c04edd25cc50f703b49e |issue=7}}
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| </ref> This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances.
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| == See also ==
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| * [[Stevens' power law]]
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| * [[Sone]]
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| * [[Nervous System]]
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| * [[Human nature]]
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| * [[Ricco's Law]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| *{{Wikisource-inline|list=
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| **{{Cite Americana|wstitle=Weber's Law|short=x |noicon=x}}
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| **{{Cite EB1911|wstitle=Weber's Law|short=x |noicon=x}}
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| }}
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| {{DEFAULTSORT:Weber-Fechner law}}
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| [[Category:Perception]]
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| [[Category:Behavioral concepts]]
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| [[Category:Psychophysics]]
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