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 physical stimulus in a quantitative fashion. Weber's law states that the just-noticeable difference between two stimuli is proportional to the magnitude of the stimuli, or, an increment is judged relative to the previous amount. 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 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 log10 (actual intensity measured with an accurate nonhuman instrument)
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 
Derivation of Fechner's law for weight perception
|This section needs additional citations for verification. (February 2011)|
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.
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 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. (1977) 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 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.
"Near miss" of Weber's law in visual regularity perception
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) showed for Glass patterns, and as van der Helm (2010) 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.
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.
- 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.
- Pinker, Steven (2011). The Better Angels Of Our Nature. P.265: Penguin. ISBN 978-0-141-03464-5.
- 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. doi:10.1017/s0140525x00029277.
- 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.
- 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.
- 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.
- van der Helm, P. A. (2010). Weber-Fechner behaviour in symmetry perception? Attention, Perception, & Psychophysics, 72, 1854-1864.
- 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.