# Weber-Fechner law

The Weber–Fechner law[1] is a proposed relationship between the magnitude of a physical stimulus and the intensity or strength that people feel.

## The case of weight

In one of his classic experiments, Weber gradually increased the weight that a blindfolded man was holding and asked him to respond when he first felt the increase. Weber found that the response was proportional to a relative increase in the weight. That is to say, if the weight is 1 kg, an increase of a few grams will not be noticed. Rather, when the mass is increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold is also doubled.[2]

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.

This logarithmic relationship is valid, not just for the sensation of weight, but for other stimuli and our sensory perceptions as well.

## References

1. 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. Gustav Theodor Fechner (1801–1887) later offered an elaborate theoretical interpretation of Weber's findings, which he called simply Weber's law, though his admirers made the law's name a hyphenate.
2. This kind of relationship can be described by a differential equation as,
$dp = k\frac{dS}{S} , \,\!$
where dp is the differential change in perception, dS is the differential increase in the stimulus and S is the stimulus at the instant. A constant factor k is to be determined experimentally. Integrating the above equation gives
$p = k \ln{S} + C, \,\!$
where $C$ is the constant of integration, ln is the natural logarithm. To determine $C$, put $p = 0$, i.e. no perception; then
$C = -k\ln{S_0}, \,\!$
where $S_0$ is that threshold of stimulus below which it is not perceived at all. Therefore, our equation becomes
$p = k \ln{\frac{S}{S_0}}. \,\!$