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 ΔI = 0.1 kg. For the weight of magnitude I = 5.0 kg, the jnd ΔI = 0.5 kg. The ratio of ΔI/I in both cases is constant (0.1).
Weber’s Law states that ΔI = 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.