Weber's Law

The psychologist Weber was responsible for discovering one of the first and most profound relationships in perceptual psychology: known as Weber's Law, it states that,

"equal relative increments of stimuli are proportional to equal increments of sensation."

That's to say that, if the intensity of a sensation (of sight, or smell, or sound, or taste, or touch) is doubled from an initial value and the sensation noted, the intensity will have to be doubled once more to give the same impression of increase in sensation. This fact leads to some startling conclusions. For example, suppose you discover a room lit with one single candle and you, finding it too dim, light a further nine candles. You find an improvement but not enough for your needs. In order to get the same improvement again you need, not 20 candles, but one hundred candles because,

And for the same improvement again,

We can re-write this relationship in terms of brightness (a perception or sensation) against intensity (number of physical candles) like this,

            Brightness         Candles


            0                      1          or         10^0

            1                      10        or         10^1

            2                      100      or         10^2

            3                      1000    or         10^3

If we let brightness be represented by b, and the number of candles by c, we can express this relationship like this,

b = log c

In words, our experience of a sensation is related to the logarithm of the stimulus.

Fascinatingly, the relationship enshrined in Weber's Law appears to apply to all the senses; touch, sight, hearing, smell and taste and it has even been suggested that our perception of time appears logarithmic, so that our perception of time is foreshortened the farther away it is from the present. Weber was particularly interested in dicovering the smallest perceptable increment of sensation because this too would always be consistent, irrespective of the intensity of the sensation. The modern term for this smallest perceptable incrememnt of sensation is, Just Noticable Difference or JND. In mathematical terms we can describe JND thus,

JND = dI/I

where I is the intensity of the stimulation.

Some interesting human-sense JNDs are tabultaed below:

Notice how our sensitivity to pitch is 70 times more accute than our sense of taste (a fact worth noting next time an acquaintance starts waxing lyrical about wine!)

Weber's Law was refined by Fechner who realsied that it would be possible to codify and express all sensation on similar, logarithmic scales if the intensity of any sensation was expressed as a ratio in realtion to the minimum threshold of a sensation. By this term is meant that there exists, for all senses, a lower limit below which nothing can be perceived, this being expressed as the threshold of vision, the threshold of taste and so on. Here are the thresholds for the five human senses:

On a practical level the logarithmic relationship between sensation and stimulus means that, when we want to express a sensation of, for example the loudness of a sound, it needs to be expressed as the logarithm of a ratio. This is the role of the decibel.


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© Richard Brice 2005. All rights reserved.