Weber–Fechner law
This includes stimuli to all senses: vision, hearing, taste, touch, and smell.
This publication was the first work ever in this field, and where Fechner coined the term psychophysics to describe the interdisciplinary study of how humans perceive physical magnitudes.
[2] He made the claim that "...psycho-physics is an exact doctrine of the relation of function or dependence between body and soul.
"[3] Ernst Heinrich Weber (1795–1878) was one of the first persons to approach the study of the human response to a physical stimulus in a quantitative fashion.
As stated above, the JND dS is proportional to the initial stimuli intensity S. Mathematically, it can be described as
It was Fechner who formulated this statement as a mathematical expression referred to as Weber contrast.
For example, the ability to perceive differences in light intensity could be related to how good that individual's vision is.
[2] He also noted that how the human sensitivity to stimuli changes depends on which sense is affected.
He used this to formulate another version of Weber's law that he named die Maßformel, the "measurement formula".
Fechner's law states that the subjective sensation is proportional to the logarithm of the stimulus intensity.
According to this law, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to logarithm of the actual intensity measured with an accurate nonhuman instrument.
For example, if a stimulus is tripled in strength (i.e., 3 × 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 × 3 × 1), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1).
The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law.
[6] An early reference to "Fechner's ... law" was in 1875 by Ludimar Hermann in Elements of Human Physiology.
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.
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.
[17] Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.
[18][19] Perception of Glass patterns[20] 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)[21] showed for Glass patterns, and as van der Helm (2010)[22] 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.
In order to be just visible, the target must be brighter or fainter than the background by some small amount
This can be seen in data collected by Blackwell[23] and plotted by Crumey,[24] showing threshold increment log
≲ 10− 5 cd m−2, approximately 25 mag arcsec−2)[24] the curves are flat - this is where the only visual perception is the observer's own neural noise ('dark light').
Obviously, this increases the dynamic range of a neuronal population, while stimulus-derived changes remain small and linear proportional.
Psychological studies show that it becomes increasingly difficult to discriminate between two numbers as the difference between them decreases.
It may also play a role in explaining why consumers neglect to shop around to save a small percentage on a large purchase, but will shop around to save a large percentage on a small purchase which represents a much smaller absolute dollar amount.
[29] It has been hypothesized that dose-response relationships can follow Weber's Law[30] which suggests this law – which is often applied at the sensory level – originates from underlying chemoreceptor responses to cellular signaling dose relationships within the body.
Dose response can be related to the Hill equation, which is closer to a power law.
Election after election, voters demand more public goods to be effectively impressed; therefore, politicians try to increase the magnitude of this "signal" of competence – the size and composition of public expenditures – in order to collect more votes.
