Weber's Law

In brief

Ernst Heinrich Weber (1795 to 1878), an anatomist and physiologist at Leipzig, noticed in the 1830s that whether we can tell two weights apart does not depend on the raw difference between them but on their ratio. Add a gram to a light weight and you feel the change; add the same gram to a heavy one and you feel nothing. The smallest change you can reliably detect, the just-noticeable-difference, is a roughly constant fraction of the starting stimulus. That regularity is Weber's law. A generation later Gustav Theodor Fechner (1801 to 1887) built a second law on top of it: if each just-noticeable-difference is treated as a fixed step of felt sensation, then sensation grows as the logarithm of stimulus intensity. With that move Fechner claimed to have measured the mind, and in 1860 he named the new field psychophysics.

The full treatment

The problem it answers

Before Fechner, the inner world of sensation seemed closed to measurement. You could measure the physical stimulus (the weight in grams, the light in candelas, the tone in decibels), but the sensation it produced, how heavy the weight feels, was private and apparently immeasurable. Philosophers from Kant onward had doubted that psychology could ever be a quantitative science for exactly this reason: there was no ruler for the subjective. Weber's ratio finding, and Fechner's logarithmic elaboration, offered the first credible answer. You cannot ask a person to report the absolute magnitude of a sensation in units, but you can ask whether two stimuli feel different, and you can find the threshold at which they begin to. From that threshold, Fechner argued, a scale of sensation could be constructed indirectly.

How Weber's law works

Weber's central quantity is the difference threshold, the smallest increase in a stimulus that a person can just detect, written as delta-I. His finding is that this threshold is proportional to the baseline intensity I:

delta-I divided by I equals k

The ratio k, now called the Weber fraction, is a constant characteristic of each sense. Lift a 100 gram weight and you may just notice a 2 gram addition; lift a 200 gram weight and you now need about 4 grams. The perceptible difference scales with the load. Weber found the fraction for lifted weight to be roughly one fortieth. Different senses have different fractions: brightness discrimination is fairly acute, loudness less so, and taste and smell cruder still. The law holds well across the middle range of intensities and breaks down at the extremes, at stimuli near the absolute threshold of detection and at very intense ones.

Fechner's law built on top

Fechner took Weber's ratio and turned it into a scale of sensation. His reasoning: treat every just-noticeable-difference as a subjectively equal unit, one step of felt sensation, no matter where on the physical scale it falls. Since each such step requires multiplying the stimulus by a constant factor rather than adding a constant amount, sensation accumulates as a logarithm of stimulus intensity:

S equals k times the logarithm of I

Here S is the felt magnitude and I the physical intensity. The consequence is the compression everyone has felt: a candle lit in a dark room is dramatic, the same candle added to a bright room is nothing; the tenth violin in an orchestra adds far less to loudness than the first. Fechner published this in Elemente der Psychophysik (Elements of Psychophysics, 1860), the founding book of the field.

The key demonstrations

Fechner did more than theorize; he invented the experimental methods that psychology still uses to find thresholds. Three of them anchor the discipline. In the method of limits, a stimulus is raised or lowered in small steps until the observer's report flips, marking the threshold. In the method of constant stimuli, fixed intensities are presented in random order many times and the threshold is read off the proportion of "yes, I detect it" responses. In the method of adjustment, the observer moves a stimulus until it just matches or just disappears. Weber's own evidence came from weight-lifting and from the two-point threshold on the skin, the smallest separation at which two touched points feel like two rather than one, which varies enormously across the body and is still used in clinical neurology.

Related distinctions

Two limits bound the enterprise. The absolute threshold is the faintest stimulus detectable at all; the difference threshold is the smallest detectable change from a baseline. Weber's law concerns the second. It is also worth separating the two laws that carry these names. Weber's law is an empirical claim about discrimination that stays close to the data. Fechner's law is a theoretical construction that adds a strong assumption: that all just-noticeable-differences are subjectively equal. That extra assumption is where later critics found their opening.

Lineage

Weber worked within the golden age of German sensory physiology, alongside Hermann von Helmholtz and Johannes Müller, treating perception as a bodily process open to measurement. Fechner, trained as a physicist and long fascinated by the relation of mind and matter, made the philosophical leap. The two laws fed directly into the birth of experimental psychology: Wilhelm Wundt, who founded the first psychology laboratory at Leipzig in 1879, absorbed the psychophysics of Weber and Fechner and took their central lesson, that mental events obey measurable laws, as license for a new science. The threshold methods Fechner devised became standard laboratory technique and remain in use today in vision science, audiology, and signal detection research.

The strongest case for it

Weber's law is one of the most robust empirical regularities in all of psychology. It holds, at least approximately, across the middle range of nearly every sensory dimension ever tested, in humans and in animals, which is why it functions as a genuine law rather than a local finding. Its deeper vindication is that ratio-based coding turns up beyond sensation: numerical estimation and judgments of time and probability show the same compression, suggesting the brain represents magnitude on a scale where equal ratios feel like equal steps. And Fechner's larger claim, that the mind can be measured indirectly through its thresholds, was vindicated by history. The methods he built in 1860 opened the door to psychophysics and quantitative perception, and they still work.

The strongest case against it

The empirical law is secure; Fechner's interpretation of it is not. The central objection came from the American psychologist S. S. Stevens (1906 to 1973), who argued in the 1950s that Fechner had guessed wrong about the shape of the sensation scale. Rather than infer sensation indirectly from stacked just-noticeable-differences, Stevens asked observers to assign numbers directly to how intense a stimulus felt, a technique called magnitude estimation. The results fit not a logarithmic law but a power law: sensation grows as stimulus intensity raised to an exponent that differs by modality. For brightness the exponent compresses, as Fechner would expect, but for electric shock it expands, so that felt intensity rises faster than the current, which a logarithm can never produce. Stevens's power law (S equals k times I raised to the power a) covers cases Fechner's cannot.

The deeper charge is that Fechner's derivation rests on an unproven assumption: that all just-noticeable-differences are subjectively equal. There is no independent reason to believe a barely detectable change feels the same size at low and high intensities, and Stevens's data suggest it does not. Weber's law itself also fails at the boundaries, deteriorating near the absolute threshold and at very high intensities, so k is not truly constant. Modern accounts often replace it with the Weber-Fechner relation as an approximation and treat signal detection theory, which separates genuine sensitivity from the observer's willingness to say "yes," as the more rigorous successor to the classical threshold concept.

Where it stands now

Both laws survive, in demoted but respectable roles. Weber's law is taught as a foundational regularity and used as a working approximation, understood to hold in the middle range and to fail at the extremes. Fechner's logarithmic law and Stevens's power law are presented side by side, the logarithm as the historic first attempt and the power law as the better empirical fit for many modalities, with the argument between them never fully resolved. What is not in dispute is Fechner's founding gesture. Psychophysics is a living field, decibels and other logarithmic scales are built into engineering, and the discovery that perception codes ratios rather than absolute differences has proved to be one of the durable facts about how minds meet the world. The measurement of sensation, once thought impossible, became ordinary.

Test yourself

Turn up the volume on a quiet song and the change is obvious; make the same adjustment on a loud one and you can barely tell. Notice that the physical increase in sound energy was identical both times. What changed was not the stimulus but its ratio to where you started. Ask whether the things you find satisfying, a raise, a faster machine, a brighter screen, obey the same rule: that each further gain has to be larger than the last to feel like anything at all.