Mathematical PsychologyWeber–Fechner law

Dr Pratyush Chaudhuri

Nirmal Hospitals and clinics

• The Weber–Fechner law attempts to describe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli.

• 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.

The case of weight

• smallest noticeable difference in weight (the least difference that the test person can still perceive as a difference), was proportional to the starting value of the weight.

• This kind of relationship can be described by a differential equation as,

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

where C is the constant of integration, ln is the natural logarithm.

• To determine C, put p = 0, i.e. no perception; then subtract − klnS0 from both sides and rearrange:•

• where S0 is that threshold of stimulus below which it is not perceived at all.

• Substituting this value in for C 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).

The case of vision

• The eye senses brightness approximately logarithmically over a fairly broad range.

• Hence stellar magnitude is measured on a logarithmic scale.

• This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C.

The case of sound

• Another logarithmic scale is the decibel scale of sound intensity.

• In the case of perception of pitch, humans hear pitch in a logarithmic or geometric ratio-based fashion.

• For notes spaced equally apart to the human ear, the frequencies are related by a multiplicative factor.

• Notation and theory about music often refers to pitch intervals in an additive way, which makes sense if one considers the logarithms of the frequencies, as

The case of numerical cognition

• Psychological studies show that numbers are thought of as existing along a mental number line.

• Larger entries are on the right and smaller entries on the left.

• 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.

Thank You.