Weber–Fechner Law

Weber's law and the just-noticeable difference

In experiments on weight and touch in 1834, German physiologist Ernst Heinrich Weber found that the smallest change needed to detect a difference between two stimuli, the "just-noticeable difference" (JND), is proportional to the size of the original stimulus. If someone holding 100 g notices a difference at 105 g, someone holding 1,000 g needs about 1,050 g. This ratio (ΔI/I = k) is the "Weber fraction," and it differs by sense: roughly 2 to 5 percent for weight, about 8 percent for brightness and under 1 percent for pitch. The law shows that we perceive the world in relative ratios rather than absolute differences. It is also known to break down for very weak or very intense stimuli (Gescheider, 1997).

Fechner's logarithmic law and psychophysics

In Elements of Psychophysics (1860), Gustav Theodor Fechner extended Weber's finding mathematically. Treating the JND as a basic unit of sensation and summing such units yields "Fechner's law": sensation magnitude S is proportional to the logarithm of stimulus intensity I (S = k log(I/I₀)). As a stimulus grows tenfold and a hundredfold, sensation increases only by one step and two steps. This is why logarithmic scales, such as decibels, stellar magnitudes and earthquake magnitudes, fit human perception well. Fechner sought to quantify the relation between mind and matter and systematized measurement methods such as the method of limits, constant stimuli and adjustment, laying the foundation of experimental psychology. Later, Thurstone's (1927) law of comparative judgment and Green and Swets's (1966) signal detection theory separated response bias from sensory judgment, refining measurement methods.

Stevens's critique and modern applications

Using magnitude estimation, in which people assign numbers directly to sensation strength, S. S. Stevens (1957, 1961) proposed "Stevens's power law," in which sensation is proportional to a power of the stimulus (S = kIⁿ), urging psychologists "to honor Fechner and repeal his law." The exponent varies by sense, about 0.33 for brightness and 3.5 for electric shock, making it an instance of a power law. Laming (1997) and others argued that the difference between the two laws stems from differences in measurement method. Meanwhile, the logarithmic law has regained attention in numerical cognition. Nieder and Miller (2003) showed that monkey prefrontal neurons represent numbers in a logarithmically compressed way, which Dehaene (2003) described as a "logarithmic mental number line." Siegler and Opfer (2003) found that children first place numbers logarithmically and shift to linear placement with learning, and Dehaene et al. (2008) reported logarithmic intuitions among Amazonian indigenous people. In economics, Daniel Bernoulli's logarithmic utility and the diminishing sensitivity to gains and losses in Kahneman and Tversky's (1979) prospect theory follow the same principle, as does price perception in which a 5-dollar discount feels large on a 10-dollar item but small on a 1,000-dollar one.

FAQ

What is Weber's law in simple terms?

A change is noticed only if it is proportional to the original amount. You quickly notice one book added to a light bag, but the same book added to a heavy bag is hard to detect.

How does Fechner's law differ from Stevens's law?

Fechner held that sensation is proportional to the logarithm of the stimulus, while Stevens, using direct magnitude estimation, argued that sensation is proportional to a power of the stimulus.

Sources reviewed: 2026-10-26

References

1. Weber, E. H. (1834). De Pulsu, Resorptione, Auditu et Tactu: Annotationes Anatomicae et Physiologicae. Leipzig: Koehler.
2. Fechner, G. T. (1860). Elemente der Psychophysik. Leipzig: Breitkopf und Härtel.
3. Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64(3), 153–181.
4. Stevens, S. S. (1961). To honor Fechner and repeal his law. Science, 133(3446), 80–86.
5. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34(4), 273–286.
6. Green, D. M., & Swets, J. A. (1966). Signal Detection Theory and Psychophysics. New York: Wiley.
7. Dehaene, S. (2003). The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences, 7(4), 145–147.
8. Nieder, A., & Miller, E. K. (2003). Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37(1), 149–157.
9. Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–243.
10. Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320(5880), 1217–1220.
11. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
12. Gescheider, G. A. (1997). Psychophysics: The Fundamentals (3rd ed.). Mahwah, NJ: Lawrence Erlbaum.
13. Laming, D. (1997). The Measurement of Sensation. Oxford: Oxford University Press.