Weber-Fechner Sensation Calculator

Weber and Fechner established the mathematical relationship between physical stimuli and perceived sensation — the foundation of psychophysics.

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. If you can just detect a 1-gram weight added to a 100-gram load, you need about a 10-gram addition to detect a change in a 1,000-gram load (k = 0.01 for lifted weights). Weber fractions for common senses: brightness ~0.02, loudness ~0.04, weight ~0.02, temperature ~0.15, taste ~0.20.

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. 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 background luminance is 1,000x 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 — a small increase in painful stimulus is felt as a large change).