Hold a 100 gram weight in one hand and a 102 gram weight in the other, and most people feel no difference. Hold a 1 kilo weight in one hand and a 1.02 kilo weight in the other and it is just as hard. The difference in grams has grown twenty times, yet it is no easier to feel. This is Weber's law, one of the oldest and most reliable findings in psychology.
The idea
In the 1830s the German physiologist Ernst Heinrich Weber studied how small a change in a stimulus has to be before it is noticed. The smallest noticeable change is called the just noticeable difference, abbreviated JND. Weber found that the JND is not a fixed amount but a fixed fraction of the starting value. If you can just notice 2 grams at 100 grams, you need about 20 grams at 1000 grams.
The fraction differs per sense. Commonly cited values, which vary with the conditions of the experiment, are around 2 percent for the weight of lifted objects, several percent for the length of a line and around 8 percent for brightness. Pitch is far finer: trained listeners can tell a difference of well under one percent in frequency. The exact numbers matter less than the pattern: in every sense, the sensitivity is relative.
From Weber to Fechner to Stevens
Gustav Fechner, a student of this field, formalised the idea in 1860. If each JND is a step of equal subjective size and each step is a fixed percentage, then the felt intensity grows with the logarithm of the physical intensity. This is the Weber–Fechner law. It explains why a doubling always feels like the same step, whether it is from 10 to 20 or from 1000 to 2000, and why sound is measured in decibels and star brightness in magnitudes.
In the 1950s S. S. Stevens measured the felt intensity directly, by asking people to assign numbers to what they sensed, and found that a power law often fits better than a logarithm. In Stevens' results apparent length grows nearly in proportion to real length, brightness grows more slowly than the physical energy and the pain of an electric shock grows much faster. The debate on the exact shape of the curve is still open, but both laws agree on the main point: perception is compressive for most senses, and what counts is the ratio.
Why it matters for estimation
- Errors are relative. A good estimator is off by a percentage, not by a fixed amount. That is why our games score most answers by ratio rather than by absolute difference.
- Log sliders. When the possible answers range from 0.2 to 20 seconds or from 1 gram to 1000 kilos, a linear slider would make the small values impossible to set. A logarithmic slider gives each doubling the same room.
- Numbers feel logarithmic, too. Children and people from cultures with little formal schooling place numbers on a line with the small ones spread out and the big ones squeezed together. Schooling gradually makes the mental number line more linear, though under time pressure the compression comes back for large numbers.
Practical consequences
Weber's law explains many ordinary experiences. One extra candle changes a dark room a lot and a bright room hardly at all. A price that rises by one euro matters on a coffee and not on a car. Two heavy suitcases feel equally heavy until the difference is several kilos. If you want to see a small difference, make the baseline smaller or compare directly: side by side, you can often detect a smaller difference than in separate judgements.
It also tells you what a reasonable goal is. Nobody can estimate a weight of 5 kilos to the gram, and nobody should try. Within a few percent is excellent for most senses, and ten percent is usual. Our scoring is set so that these accuracies earn good scores and gross errors earn few.
Try it
The games below make Weber's law tangible. In the line and dot games you can feel how the task gets harder as the two sides become closer in ratio, regardless of how big they are.