Weber's law

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. Fechner was a student of Weber and named his first law in honor of his mentor, since it was Weber who had conducted the experiments needed to formulate the law.^[4]

Fechner formulated several versions of the law, all communicating the same idea. One formulation states:

Simple differential sensitivity is inversely proportional to the size of the components of the difference; relative differential sensitivity remains the same regardless of size.^[2]

What this means is that the perceived change in stimuli is proportional to the initial stimuli.

Weber's law also incorporates the just-noticeable difference (JND). This is the smallest change in stimuli that can be perceived. As stated above, the JND dS is proportional to the initial stimuli intensity S. Mathematically, it can be described as ${\displaystyle dS=K\cdot S}$ where ${\displaystyle S}$ is the reference stimulus and ${\displaystyle K}$ is a constant.^[5] It may be written as Ψ = k logS, with Ψ being the sensation, ${\displaystyle k}$ being a constant, and ${\displaystyle S}$ being the physical intensity of the stimulus.

Weber's law always fails at low intensities, near and below the absolute detection threshold, and often also at high intensities, but may be approximately true across a wide middle range of intensities.^[6]

Weber contrast

Although Weber's law includes a statement of the proportionality of a perceived change to initial stimuli, Weber only refers to this as a rule of thumb regarding human perception. It was Fechner who formulated this statement as a mathematical expression referred to as Weber contrast.^[2][7][8][9]

$${\displaystyle dp={\frac {dS}{S}}\,\!}$$

Weber contrast is not part of Weber's law.^[2][7]

Fechner's law

Fechner noticed in his own studies that different individuals have different sensitivity to certain stimuli. 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.^[7]

$${\displaystyle p=k\ln {\frac {S}{S_{0}}}\,\!}$$

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 × 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). 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.^[10][11]

Since Weber's law fails at low intensity, so does Fechner's law.^[6]

An early reference to "Fechner's ... law" was in 1875 by Ludimar Hermann in Elements of Human Physiology.^[12]

Deriving Fechner's law

Fechner's law is a mathematical derivation of Weber contrast.

$${\displaystyle dp=k{\frac {dS}{S}}}$$

Integrating the mathematical expression for Weber contrast gives:

$${\displaystyle p=k\ln {S}+C}$$

where ${\displaystyle C}$ is a constant of integration and ln is the natural logarithm.

To solve for ${\displaystyle C}$, assume that the perceived stimulus becomes zero at some threshold stimulus ${\displaystyle S_{0}}$. Using this as a constraint, set ${\displaystyle p=0}$ and ${\displaystyle S=S_{0}}$. This gives:

$${\displaystyle C=-k\ln {S_{0}}}$$

Substituting ${\displaystyle C}$ in the integrated expression for Weber's law, the expression can be written as:

$${\displaystyle p=k\ln {\frac {S}{S_{0}}}}$$

The constant k is sense-specific and must be determined depending on the sense and type of stimulus.^[7]

Weight perception

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. 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.

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.^[13][14] Following the work of S. S. Stevens, many researchers came to believe in the 1960s that the Stevens's power law was a more general psychophysical principle than Fechner's logarithmic law.

Sound

Weber's law does not quite hold for loudness. It is a fair approximation for higher intensities, but not for lower amplitudes.^[15]

Vision

Uneven light from a window.

The same light, redistributed by prism tiles in the window.

Because of the ~logarithmic perception of light levels, if light is redistributed from the brightest parts of a room to the dimmest, the room appears brighter overall, and more space can be given a useful and comfortable level of illumination.

The eye senses brightness approximately logarithmically over a moderate range and stellar magnitude is measured on a logarithmic scale.^[17] 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.^[17] Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.^[18][19]

Limitation of Weber's law at low light levels

Threshold increment versus background luminance for various target diameters (in arcmin). Data from tables 4 and 8 of Blackwell (1946), plotted in Crumey (2014).

For vision, Weber's law implies constancy of luminance contrast. Suppose a target object is set against a background luminance ${\displaystyle B}$. In order to be just visible, the target must be brighter or fainter than the background by some small amount ${\displaystyle \Delta B}$. The Weber contrast is defined as ${\displaystyle C=\Delta B/B}$, and Weber's law says that ${\displaystyle C}$ should be constant for all ${\displaystyle B}$.

Human vision follows Weber's law closely at normal daylight levels (i.e. in the photopic range) but begins to break down at twilight levels (the mesopic range) and is completely inapplicable at low light levels (scotopic vision). This can be seen in data collected by Blackwell^[23] and plotted by Crumey,^[24] showing threshold increment log${\displaystyle \Delta B}$ versus background luminance log${\displaystyle B}$ for various targets sizes. At daylight levels, the curves are approximately straight with slope 1, i.e. log${\displaystyle \Delta B}$ = log${\displaystyle B+const.}$, implying ${\displaystyle C=\Delta B/B}$ is constant. At the very darkest background levels (${\displaystyle B}$ ≲ 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'). In the intermediate range, a portion can be approximated by the De Vries - Rose law, related to Ricco's law.

Lognormal distributions

Activation of neurons by sensory stimuli in many parts of the brain is by a proportional law: neurons change their spike rate by about 10–30%, when a stimulus (e.g. a natural scene for vision) has been applied. However, as Scheler (2017)^[25] showed, the population distribution of the intrinsic excitability or gain of a neuron is a heavy tail distribution, more precisely a lognormal shape, which is equivalent to a logarithmic coding scheme. Neurons may therefore spike with 5–10 fold different mean rates. Obviously, this increases the dynamic range of a neuronal population, while stimulus-derived changes remain small and linear proportional.

An analysis^[26] of the length of comments in internet discussion boards across several languages shows that comment lengths obey the lognormal distribution with great precision. The authors explain the distribution as a manifestation of the Weber–Fechner law.

Numerical cognition

Psychological studies show that it becomes increasingly difficult to discriminate between two numbers as the difference between them decreases. This is called the distance effect.^[27][28] This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances. 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]

Pharmacology

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.

Public finance

There is a new branch of the literature on public finance hypothesizing that the Weber–Fechner law can explain the increasing levels of public expenditures in mature democracies. 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.^[31]

Emotion

Preliminary research has found that pleasant emotions adhere to Weber’s Law, with accuracy in judging their intensity decreasing as pleasantness increases. However, this pattern wasn't observed for unpleasant emotions, suggesting a survival-related need for accurately discerning high-intensity negative emotions.^[32]