Weber's Law

Background:

Weber's law is often written as:

ΔI / I = k

where Δ mean "change of", I means intensity and k is a constant (remember that Fechner and Weber were German, and in German, constant is spelled konstante.) ΔI should be read as "the change in intensity needed to just notice a difference" and is sometimes written as JND for "just noticeable difference." Putting it all together, Weber's law states that the change in intensity needed to just notice a change in intensity divided by the original intensity is always the same value, k, for a given sensory modality.

A slight algebraic manipulation and substitution of equivalent terms in the above formula yields:

JND = kI

This alternative says the same thing, but in a different way: the change in intensity needed to just notice a difference (JND) is proportional (k) to the original intensity (I). This implies that low intensity stimuli (e.g. a dim light, a soft sound) require very small changes in their intensity in order to be just noticeably different, but that high intensity stimuli (e.g. a bright light, a loud sound) require a much larger change in their intensity to be just noticeably different.

You might have experienced something like this with a 3-way light bulb. A 3-way light bulb has three settings (in addition to "off".) For example, it might have a 50 W, 100 W and 150 W settings. If we assume that the 100 W setting produces twice the amount of light as the 50 W setting, then you can easily see Weber's law in action. Changing the bulb from 50 to 100 W makes a large, easily noticeable difference in the perceived brightness of the light. However, changing the bulb from 100 to 150 W makes a much smaller, not so easily noticed difference in the perceived brightness of the light. This is what Weber's law predicts -- the larger original intensity (100 W) requires a much larger change in intensity to be just noticeably different than the smaller original intenstiy (50 W.) Alternatively, the same increase in the intensity of the lamp (50 to 100 W vs 100 to 150 W) will be more noticeable at the lower intenstiy than at the higher intensity.

You could use a psychophysical procedure, such as Fechner's method of adjustment to determine the Weber fraction for a given sensory modality. For example, you could measure the intensity of a sound. For example, the sound's intensity might be 100 (I am intentionally leaving the unit of measurement off -- it doesn't really matter what the unit is as long as you consistently use the same unit throughout the example. The units will cancel with each other, leaving Weber's fraction as a dimensionless measurement.) Then you would gradually increase the intensity of sound until you could just notice that it was just louder than it originally was. In this example, the just noticeably louder sound has an intensity of 104.8. From those two numbers, you could calculate the Weber fraction:

ΔI / I = k
(104.8 - 100) / 100 = 0.048

ΔI is the change in intensity needed to just notice the differences -- that is the difference of the louder and softer sounds: 104.8 - 100. The original intensity, I, is 100. Doing the math yields a Weber fraction, k, equal to .048

Now that we have the Weber fraction, we can use it to judge how much louder a sound must be than any other sound in order to just notice that it is louder. For example, how much louder does a sound have to be in order to be noticeably louder than a sound with an intensity of 50? The change in intensity to be just noticeably different is:

JND = kI
JND = 0.048 x 50 = 2.4

We need to add that value to the original intensity (I) to get the intensity that would be just noticeably different. 50 + 2.4 = 52.4. Thus, the person should be able to just discriminate the loudness of sounds with intensities of 50 and 52.4.

We could easily do the last two steps in a single step:

Intensity at which the difference can be just noticed = I * (k + 1)
= 50 * (0.048 + 1)
= 52.4

To check your work, you could calculate the Weber fraction, k, from these two values. It should still equal 0.048.

ΔI / I = k
(52.4 - 50) / 50 = 0.048