In 1834, the psychophysicist E. H. Weber formulated one of the most fundamental insights into human perception, known as Weber’s Law. This states that there is usually a relationship between the quantity of something and how much more must be added to it for us to be able to perceive that an increase has taken place.

If you hold an object weighing 1.0 kg and then hold another object weighing 1.05 kg you may not notice any difference in their weights, but you do notice that something weighing 1.1 kg is heavier. However, if you start with a 5.0 kg object, it would take 0.5 kg to be added to the weight until you notice the difference. In other words, the ‘just noticeable difference’ (jnd) changes depending on what the starting quantity is.

In this example, for the weight of magnitude *I* = 1.0kg, the jnd was Δ*I* = 0.1 kg. For the weight of magnitude *I* = 5.0 kg, the jnd Δ*I* = 0.5 kg. The ratio of Δ*I*/*I* in both cases is constant (0.1).

Weber’s Law states that Δ*I* = K *I ,* where K is a constant called the Weber Fraction (often expressed as a percentage), which is particular to the physical property. A Weber fraction of 1% means a high sensitivity to increments in the property, but a fraction of 20% indicates a much lower sensitivity to increments.

Weber’s Law applies to the property of length, which is perhaps the most common geometric property used to encode numerical values. We might expect to be able to decode lengths fairly well, but this example from Bill Cleveland’s book “The Elements of Graphing Data” shows that this is not always the case.

It is very difﬁcult to tell that the lengths of A and B are different.

But if we add a simple “scale” to the bars by enclosing them in frames of equal heights and with tickmarks half way up the sides of the frames, it becomes much easier to intuitively decode and compare the lengths.

This works because we can now also compare the lengths of the white bars (as well as the distances of the tops of the black bars from the tickmarks).

The lengths of the white portions of the bars differ by the same absolute amount as the lengths of the black portions. The fact that the white bars are clearly of different lengths while the black ones are not makes it obvious that it is the relative difference that we perceive rather than the absolute one.

Interesting post, Andrew.

This also got me thinking: if those bars are shown at the same time but in different places (particularly not lined up) it is hard to see that the black bars differ in length, but if they were shown in the same place but at different times – i.e., an animation between the two – I imagine the difference would jump out straight away.

Does Weber’s Law apply to a *rate of change* rather than just a change in this case? We’re all familiar with the experience that something changing very slowly is much harder to spot than something changing quickly, and this works in combination with the proportional factor of Weber. So a weight that that became 10% heavier over a lengthy period would be less easy to spot than one that suddenly gained that much extra weight.

But I’m sure you already have ideas to cover the brain’s ability to pick out “temporal changes” in a future post!