The Weber–Fechner law combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law. Ernst Heinrich Weber (1795–1878) was one of the first people to approach the study of the human response to a physical stimulus in a quantitative fashion. His law states that the just-noticeable difference between two stimuli is proportional to the magnitude of the stimuli. Gustav Theodor Fechner (1801–1887) later offered an elaborate theoretical interpretation of Weber's findings, in which he attempted to describe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli. Fechner's law states that subjective sensation is proportional to the logarithm of the stimulus intensity.
Derivation of Fechner's law for weight perception 
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Weber found that the just noticeable difference (jnd) 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 0.005xg 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.
This kind of relationship can be described by the differential equation
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.
Integrating the above equation gives
To solve for , put , i.e., no perception; then subtract from both sides and rearrange:
where is that threshold of stimulus below which it is not perceived at all.
Substituting this value in for above and rearranging, our equation becomes:
The relationship between stimulus and perception is 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 link1 or link2).
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. 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.
Other sense modalities provide only mixed support for either Weber's law or Fechner's law.
The case of vision 
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. 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. Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.
The case of sound 
"Near miss" of Weber's law in the auditory system 
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. in their 1976 paper 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.
The case of numerical cognition 
Psychological studies show that numbers are thought of as existing along a mental number line. 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. This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances.
See also 
- Reber, A.S.(1985) The Penguin dictionary of psychology. London: Penguin Books
- 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.
- Heidelberger, M. (2004)Nature from within: Gustav Theodor Fechner and his psychophysical worldview. Transl. C. Klohr. Pittsburgh, USA: University of Pittsburg Press.
- Masin, S.C.; Zudini, V.; Antonelli, M. (2009). "Early alternative derivations of Fechner's law". J. History of the Behavioral Sciences 45: 56–65. doi:10.1002/jhbs.20349.
- Murray, D.J. (1993). "A perspective for viewing the history of psychophysics". Behavioral and Brain Sciences 16: 115–186.
- V. B. Bhatia (2001). Astronomy and astrophysics with elements of cosmology. CRC Press. p. 20. ISBN 978-0-8493-1013-3.
- Jianhong (Jackie) Shen; Yoon-Mo Jung (2006). "Weberized Mumford–Shah model with Bose–Einstein photon noise". Appl. Math. Optim. 53 (3): 331–358. doi:10.1007/s00245-005-0850-1.
- Jianhong (Jackie) Shen (2003). "On the foundations of vision modeling I. Weber's law and Weberized TV (total variation) restoration". Physica D: Nonlinear Phenomena 175 (3/4): 241–251. doi:10.1016/S0167-2789(02)00734-0.
- Moyer R.S., Landauer T.K. (September 1967). "Time required for judgements of numerical inequality". Nature 215 (5109): 1519–20. doi:10.1038/2151519a0. PMID 6052760.
- Longo M.R., Lourenco S.F. (2007). "Spatial attention and the mental number line: evidence for characteristic biases and compression". Neuropsychologia 45 (7): 1400–6. doi:10.1016/j.neuropsychologia.2006.11.002.