Just-noticeable difference

Why does one extra candle brighten a dim room yet disappear entirely in a bright one?

The candle does not know how many others are burning. It puts out the same light in an empty room and in a lit hall, and the physicist's meter agrees: the same quantity has been added in both cases. Yet in the first room the change is unmistakable and in the second it is invisible. So the failure is not in the candle and not in the light. It is in the comparison — and that means the interesting question is not "how small a change can we detect" but "how small a change relative to what".

Suppose we ran the experiment properly. Take a standard stimulus and raise it slowly until the observer reliably says it has grown. Record that smallest detectable increment — the just-noticeable difference. Then repeat at a larger standard, and larger again. What shape should the results have?

If detection were a matter of a fixed floor — some absolute quantity below which nothing registers — the increment would be the same at every level, and one candle would either always work or never work. That is plainly not what happens, so scrap it.

The alternative is that the increment scales with the level you started from, and this is what is found across sense after sense: the just-noticeable difference is roughly a constant fraction of the standard. Double the background and you must roughly double the increment. That is Weber's law, and the candle question is answered by it directly — one added to one is an enormous proportional change, one added to a hundred is a tiny one, and only the proportion is being judged.

But naming the regularity is not explaining it. Why would a sense work in ratios? Here is the reasoning that I think carries the most weight. Detection is not reading a number off a dial; it is deciding whether the current sensation differs from the previous one by more than the sensation wobbles on its own. Every neural estimate of intensity is noisy — the same light produces a slightly different internal signal each moment. So the increment you need is whatever it takes to stand clear of that noise.

Now put the two together. If the internal noise were a fixed amount, the required increment would be fixed too, and we already ruled that out. So the noise must grow with the signal. And there is good reason for it to: neurons signal partly by firing rate, and the variability of a count of events tends to grow with the count. If variability rises roughly in proportion to intensity, then the increment that stands clear of it must rise in proportion too — and Weber's law falls straight out. The law is not a rule the senses obey; it is what proportional noise looks like from the outside.

Does that buy us a test? It does, and a sharp one. If the constant fraction comes from proportional noise, then the law must fail exactly where the noise stops being proportional — at the very bottom of the range, where a floor of spontaneous background activity dominates and the signal contributes little to the variability. There the required fraction should not stay constant; it should rise as the standard falls, because you are now fighting a fixed noise floor rather than a proportional one. If instead the fraction stayed flat all the way down to absolute threshold, the account would be wrong. What is observed is the rise. Weber's law is a middle-range approximation, and its failure at the bottom is not an embarrassment but a confirmation of why it holds in the middle.

Three refinements, and one place where the reputation of this idea runs ahead of it.

First, the constant fraction is not one number. It differs sharply between senses — fine for pitch, coarse for taste and smell — and, within a sense, between the dimensions being judged. I will not quote figures, because published values depend heavily on the method used to measure them and on what counts as "reliably detected", and a number given without that context misleads more than it informs.

Second, this is a different claim from the ones in Dark adaptation and Habituation versus sensitization, and the difference is worth fixing. Those are about the state of the detector changing over time — pigment being spent and restocked, a response declining with repetition. Weber's law holds at a fixed adapted state: nothing about the observer is changing, and the fraction is a property of the comparison, not of the detector's stock. It is also the reason a loudness scale, as in Broadcast loudness, cannot simply be linear in physical power.

Third, and this is where care is needed: a famous next step is not as solid as it sounds. If each just-noticeable difference is one step, and steps are proportional to level, then adding steps gives a logarithmic sensation scale — Fechner's law. That derivation smuggles in an assumption, that all just-noticeable differences feel subjectively equal, which was never measured and is doubtful. When people were later asked to estimate magnitudes directly, the results fitted power functions rather than a logarithm, with exponents that differ by sense — compressive for brightness, expansive for electric shock. So the discrimination law is robust and the sensation scale built on it is not; they are routinely quoted together as though equally settled.

• How signal detection theory separates sensitivity from a willingness to say "yes", and why that undermines the older idea of a threshold as a fixed point.
• Why magnitude estimation and discrimination give different scales for the same sense.

Every term the collection defines is gathered in the glossary.