Continuum	Exponent ()	Stimulus condition
Loudness	0.67	Sound pressure of 3000 Hz tone
Vibration	0.95	Amplitude of 60 Hz on finger
Vibration	0.6	Amplitude of 250 Hz on finger
Brightness	0.33	5° target in dark
Brightness	0.5	Point source Point source A point source is a localised, relatively small source of something.Point source may also refer to:*Point source , a localised source of pollution**Point source water pollution, water pollution with a localized source...
Brightness	0.5	Brief flash
Brightness	1	Point source briefly flashed
Lightness	1.2	Reflectance of gray papers
Visual length	1	Projected line
Visual area	0.7	Projected square
Redness (saturation)	1.7	Red-gray mixture
Taste	1.3	Sucrose Sucrose Sucrose is the organic compound commonly known as table sugar and sometimes called saccharose. A white, odorless, crystalline powder with a sweet taste, it is best known for its role in human nutrition. The molecule is a disaccharide composed of glucose and fructose with the molecular formula...
Taste	1.4	Salt Sodium chloride Sodium chloride, also known as salt, common salt, table salt or halite, is an inorganic compound with the formula NaCl. Sodium chloride is the salt most responsible for the salinity of the ocean and of the extracellular fluid of many multicellular organisms...
Taste	0.8	Saccharin Saccharin Saccharin is an artificial sweetener. The basic substance, benzoic sulfilimine, has effectively no food energy and is much sweeter than sucrose, but has a bitter or metallic aftertaste, especially at high concentrations...
Smell	0.6	Heptane Heptane n-Heptane is the straight-chain alkane with the chemical formula H3C5CH3 or C7H16. When used as a test fuel component in anti-knock test engines, a 100% heptane fuel is the zero point of the octane rating scale...
Cold	1	Metal contact on arm
Warmth	1.6	Metal contact on arm
Warmth	1.3	Irradiation of skin, small area
Warmth	0.7	Irradiation of skin, large area
Discomfort, cold	1.7	Whole body irradiation
Discomfort, warm	0.7	Whole body irradiation
Thermal pain	1	Radiant heat on skin
Tactual roughness	1.5	Rubbing emery cloths
Tactual hardness	0.8	Squeezing rubber
Finger span	1.3	Thickness of blocks
Pressure on palm	1.1	Static force on skin
Muscle force	1.7	Static contractions
Heaviness	1.45	Lifted weights
Viscosity	0.42	Stirring silicone fluids
Electric shock	3.5	Current through fingers
Vocal effort Vocal effort Vocal effort is a quantity varied by speakers when adjusting to an increase or decrease in the communication distance. The communication distance is the distance between the speaker and the listener. Vocal effort is a subjective physiological quantity, and is mainly dependent on subglottal...	1.1	Vocal sound pressure
Angular acceleration	1.4	5 s rotation
Duration	1.1	White noise stimuli

Stevens' power law is a proposed relationship between the magnitude

Magnitude (mathematics)

The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 of a physical stimulus and its perceived intensity or strength. It is often considered to supersede the Weber–Fechner law

Weber–Fechner law

The Weber–Fechner law is a confusing term, because it combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law...

 on the basis that it describes a wider range of sensations, although critics argue that the validity of the law is contingent on the virtue of approaches to the measurement

Measurement

Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

 of perceived intensity that are employed in relevant experiments. In addition, a distinction has been made between (i) local psychophysics

Psychophysics

Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they effect. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual...

, where stimuli are discriminated only with a certain probability, and (ii) global psychophysics, where the stimuli would be discriminated correctly with near certainty (Luce

R. Duncan Luce

Robert Duncan Luce is the Distinguished Research Professor of Cognitive Science at the University of California, Irvine.Luce received a B.S. in Aeronautical Engineering from the Massachusetts Institute of Technology in 1945, and PhD in Mathematics from the same university in 1950...

 & Krumhansl, 1988). The Weber–Fechner law and methods described by L.L. Thurstone are generally applied in local psychophysics, whereas Stevens' methods are usually applied in global psychophysics.

The theory is named after psychophysicist Stanley Smith Stevens

Stanley Smith Stevens

Stanley Smith Stevens was an American psychologist who founded Harvard's Psycho-Acoustic Laboratory and is credited with the introduction of Stevens' power law. Stevens authored a milestone textbook, the 1400+ page "Handbook of Experimental Psychology" . He was also one of the founding organizers...

 (1906–1973). Although the idea of a power law

Power law

A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

 had been suggested by 19th-century researchers, Stevens is credited with reviving the law and publishing a body of psychophysical data to support it in 1957.

The general form of the law is
where is the magnitude of the physical stimulus, ψ(I) is the psychophysical function relating to the subjective magnitude of the sensation evoked by the stimulus, a is an exponent that depends on the type of stimulation and k is a proportionality

Proportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

 constant that depends on the type of stimulation and the units used.

The table to the right lists the exponents reported by Stevens.

Methods

The principal methods used by Stevens to measure the perceived intensity of a stimulus were magnitude estimation and magnitude production. In magnitude estimation with a standard,
the experimenter presents a stimulus called a standard and assigns it a number called the modulus. For subsequent stimuli, subjects report numerically their perceived intensity relative to the standard so as to preserve the ratio between the sensations and the numerical estimates (e.g., a sound perceived twice as loud as the standard should be given a number twice the modulus). In magnitude estimation without a standard (usually just magnitude estimation), subjects are free to choose their own standard, assigning any number to the first stimulus and all subsequent ones with the only requirement being that the ratio between sensations and numbers is preserved. In magnitude production a number and a reference stimulus is given and subjects produce a stimulus that is perceived as that number times the reference. Also used is
cross-modality matching, which generally involves subjects altering the magnitude of one physical quantity, such as the brightness of a light, so that its perceived intensity is equal to
the perceived intensity of another type of quantity, such as warmth or pressure.

Criticisms

Stevens generally collected magnitude estimation data from multiple observers, averaged the data across subjects, and then fitted the data to a power function. Because the fit was generally reasonable, he concluded the power law was correct. This approach ignores any individual differences that may obtain and indeed it has been reported that the power relationship does not always hold as well when data are considered separately for individual respondents .

Another issue is that the approach provides neither a direct test of the power law itself nor the underlying assumptions of the magnitude estimation/production method.

Stevens' main assertion was that using magnitude estimations/productions respondents were able to make judgements on a ratio scale

Scale (ratio)

The scale ratio of some sort of model which represents an original proportionally is the ratio of a linear dimension of the model to the same dimension of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such...

 (i.e., if x and y are values on a given ratio scale, then there exists a constant k such that x = ky). In the context of axiomatic

Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...

 psychophysics, formulated a testable property capturing the implicit underlying assumption this assertion entailed. Specifically, for two proportions p and q, and three stimuli, x, y, z, if y is judged p times x, z is judged q times y, then t = pq times x should be equal to z. This amounts to assuming that respondents interpret numbers in a veridical way. This property was unambiguously rejected . Without assuming veridical interpretation of numbers, formulated another property that, if sustained, meant that respondents could make ratio scaled judgments, namely, if y is judged p times x, z is judged q times y, and if y is judged q times x, z is judged p times y, then z should equal z. This property has been sustained in a variety of situations .

Because Stevens fit power functions to data, his method did not provide a direct test of the power law itself. , under the condition that respondents' numerical distortion function and the psychophysical functions could be separated, formulated a behavioral condition equivalent to the psychophysical function being a power function. This condition was confirmed for just over half the respondents and the power form was found to be a reasonable approximation for the rest .

It has also been questioned, particularly in terms of signal detection theory, whether any given stimulus is actually associated with a particular and absolute perceived intensity; i.e. one that is independent of contextual factors and conditions. Consistent with this, Luce (1990, p. 73) observed that "by introducing contexts such as background noise in loudness judgements, the shape of the magnitude estimation functions certainly deviates sharply from a power function".