Weber-Fechner Law: Why Every Difference Is Judged Against a Base

A difference is never judged on its own. It is judged next to the quantity it was added to, and the larger that quantity, the larger the difference has to be before anyone notices. Suppose 2 grams added to an 80 gram envelope is the smallest increase you can feel. Weber's law puts the step at 20 grams on an 800 gram bag of flour, so the same 2 grams goes unnoticed. Nothing about the 2 grams changed; what changed is what it was weighed against.

Ernst Weber's version, published in 1834, is a statement about thresholds: the smallest detectable difference between two stimuli is close to a constant proportion of their magnitude. That proportion, the Weber fraction, differs by sense and by person, and holds best in the middle of a range rather than at the extremes. Gustav Fechner's version, published in 1860, goes one step further and describes the whole scale rather than the threshold: if equal proportional steps in the stimulus produce equal steps in sensation, then sensation grows with the logarithm of intensity. Doubling the light in a room does not double the brightness you see.

The same compression shows up in judgments that have nothing to do with the senses. A 5 dollar saving is a third of the price of a 15 dollar calculator and a twenty-fifth of the price of a 125 dollar one, and people treat it accordingly, although the saving is the same either way. In Kahneman and Tversky's prospect theory the property is called diminishing sensitivity, and it is one reason the value function flattens as amounts grow: the difference between 10 and 20 dollars does more work than the difference between 1,010 and 1,020.

The exact shape of the function is still argued over. S. S. Stevens showed in 1957 that when people are asked to rate magnitudes directly rather than detect differences, the ratings often follow a power function of intensity rather than a logarithm, with an exponent that varies by sense. The proportional threshold Weber described is the robust part. Fechner's logarithm is a good approximation in some ranges and senses and a poor one in others, so the law is best read as a description of compression rather than a formula that fits every judgment.

Ernst Heinrich Weber, an anatomist and physiologist at Leipzig, reported the proportional threshold in 1834 in a Latin monograph on touch and the senses. He had people lift pairs of weights and say which was heavier, and found that the smallest difference they could detect was not a fixed number of grams but a share of the weights lifted: a heavier pair needed a bigger difference. He reported the same proportionality for the lengths of lines judged by eye.

Gustav Theodor Fechner, also at Leipzig, published Elemente der Psychophysik in 1860 and gave the field its name. His contribution was to treat Weber's threshold as the unit of a scale: if the smallest noticeable step is always the same fraction of the current value, then counting steps from a starting point gives a measure of sensation, and that measure grows as the logarithm of the stimulus. That is the form most people mean by the Weber-Fechner law, and it was an early attempt to put a number on an experience.

The money half of the demo comes from a much later experiment. Amos Tversky and Daniel Kahneman put this problem to respondents in their 1981 paper in Science. You are about to buy a jacket for 125 dollars and a calculator for 15 dollars, and the salesman says the calculator is on sale for 10 dollars at the other branch, 20 minutes away. Would you make the trip? A second version swapped the prices, so the jacket cost 15 dollars and the calculator 125, and the sale saved the same 5 dollars. In the first version 68 percent said they would make the trip; in the second, 29 percent did. The trip and the saving are the same. What changed is the price the saving was set against.

Richard Thaler had set out the same structure in 1980 and explained it as mental accounting: the discount is booked against the purchase it came from rather than against the money itself. Reading a difference as a proportion of its base is the perceptual habit; filing it against a purchase is what people do with it.

The pattern appears wherever a number arrives next to a larger one. The rows below are illustrations, not findings from the studies cited on this page.

The compression cannot be switched off, but the base it works against can be removed from the comparison.

• Take the amount out of its context. A 500 dollar option is a 500 dollar purchase. Ask what else 500 dollars buys, not what share of the total it is.
• Write a rule in absolute terms and use it every time. A fixed number of dollars per hour of driving, or a fixed amount above which a purchase waits a day, settles the comparison before the base can bend it.
• Convert percentages to money and money to percentages, and look at both.
• Watch for small steps repeated. A difference below your threshold is still a difference, and a series of them adds up to a change you would have refused in one move.
• In sensory judgments, trust a calibrated measurement rather than a felt step. A kitchen scale reads the same 2 grams whatever the load, which is why recipes are written in grams. Check the units, though: a decibel is a ratio on a logarithmic scale, so a meter reading in decibels expresses the compression rather than removing it.

Several effects change how a quantity is judged. They differ in what the quantity is being compared with.

• It is not mental accounting, the habit of filing money into separate budgets so that a saving on a purchase is treated differently from the same amount in a bank account. The Weber-Fechner law says the saving is read against the price; mental accounting says which account the two go in.
• It is not the contrast effect, where a judgment shifts because of something presented next to it. In a contrast effect the comparison is a neighbor; here it is the magnitude of the thing being changed.
• It is not loss aversion, the finding that a loss weighs more than a gain of the same size. That is an asymmetry between directions; this is a compression of magnitudes in both directions.