Weber-Fechner Sensation Calculator

Weber and Fechner established the mathematical relationship between physical stimuli and perceived sensation, which is the foundation the whole of psychophysics is built on.

Weber’s law (Ernst Heinrich Weber, 1834):

Delta_S = k x S

The just-noticeable difference (JND) is a constant fraction k of the current stimulus S. Lifted weight has a Weber fraction of about 0.02, so if you can just detect 2 grams added to a 100-gram load, you need roughly 20 grams before you notice a change in a 1,000-gram load. The absolute amount grew tenfold; the fraction did not move.

Typical Weber fractions, from Teghtsoonian’s 1971 review:

Treat all of these as approximate. Weber fractions shift with the measurement method, the range tested and the observer, and different textbooks quote noticeably different figures for the same sense. They also break down at the extremes: near the absolute threshold, and near the point where the sense saturates, k rises sharply.

How many steps can you tell apart? Because each JND is a constant fraction rather than a constant amount, the steps get bigger as intensity rises, and the number of discriminable levels between the threshold S0 and some intensity S is logarithmic:

Steps = ln(S / S0) / ln(1 + k)

This is worth being careful about. A tempting shortcut is to divide by k, but 1/k is a fixed number that takes no account of how loud or bright the stimulus actually is, and Weber’s law is precisely the claim that it should.

Fechner’s law (Gustav Fechner, 1860):

psi = k x ln(S / S0)

Perceived sensation psi grows as the logarithm of stimulus intensity relative to the absolute threshold S0. Fechner derived this by treating every JND as one equal step of sensation and integrating Weber’s law, which is why the same k appears in both. Doubling the intensity does not double the sensation, it adds a constant increment. This explains why a phone screen seems much brighter indoors than outdoors, where the background luminance is a thousand times higher.

Stevens’ power law (S.S. Stevens, 1957):

psi = k x S^n

For most senses, the power law fits data better than the logarithmic Fechner law. Exponents n vary by modality: brightness n = 0.33 (compressed), loudness n = 0.67, vibration n = 0.95 (nearly linear), pain n = 1.5 (amplified, so a small increase in a painful stimulus is felt as a large one).