# Stevens' power law

Continuum Exponent (${\displaystyle a}$) 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
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
Taste 1.4 Salt
Taste 0.8 Saccharine
Smell 0.6 Heptane
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 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 of a physical stimulus and the intensity or strength that people feel.

Most people think that it describes a wider range of sensations than Weber-Fechner law. But critics argue that the validity of the law is not sure.

The theory is named after psychophysicist Stanley Smith Stevens (19061973). Although the idea of a power law 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 1956.

The general form of the law is

${\displaystyle \psi (I)=kI^{a},\,\!}$

where ${\displaystyle I}$ is the magnitude of the physical stimulus, ${\displaystyle \psi }$ is the psychophysical function capturing sensation (the subjective size of the stimulus), ${\displaystyle a}$ is an exponent that depends on the type of stimulation and ${\displaystyle k}$ is a proportionality constant that depends on the type of stimulation and the units used.

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

## References

• Ellermeier, W., Faulhammer, G. (2000). Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production. Perception & Psychophysics, 62, 1505–1511.
• Green, D. M., & Luce, R. D. (1974). Variability of magnitude estimates: a timing theory analysis. Perception & Psychophysics, 15, 291–300.
• Luce, R. D. (2002). A psychophysical theory of intensity proportions, joint presentations, and matches. Psychological Review, 109, 520–532.
• Narens, L. (1996). A theory of ratio magnitude estimation. Journal of Mathematical Psychology, 40, 109–129.
• Smelser, N. J., & Baltes, P. B. (2001). International encyclopedia of the social & behavioral sciences. pp. 15105–15106. Amsterdam; New York: Elsevier. ISBN 0-08-043076-7.
• Steingrimsson, R., & Luce, R. D. (2006). Empirical evaluation of a model of global psychophysical judgments: III. A form for the psychophysical function and intensity filtering. Journal of Mathematical Psychology, 50, 15–29.
• Stevens, S. S. (1957). On the psychophysical law. Psychological Review 64(3):153–181. PMID 13441853.
• Zimmer, K. (2005). Examining the validity of numerical ratios in loudness fractionation. Perception & Psychophysics, 67, 569–579.