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Weber's discovery
In the 1830s, the physiologist Ernst Heinrich Weber found that the smallest difference a person can detect is a fixed proportion of the original stimulus, not a fixed amount. Someone who can just tell a 100-gram weight from a 102-gram weight can just tell a 200-gram weight from a 204-gram one, a threshold of about 2 percent. Gustav Fechner later extended this into the idea that perceived sensation is proportional to the logarithm of the physical stimulus.
Load times
For load times, web performance specialists put the just noticeable difference at roughly 20 percent of the current duration. A page that loads in 5 seconds has to reach 4 seconds or less before users reliably see a difference, so 200 milliseconds off it is functionally imperceptible. The same 200 milliseconds off a 500-millisecond load is substantial. Going from 10 seconds to 9 is a 10 percent gain, and going from 2 seconds to 1 is a 50 percent gain.
Prices
Prices follow the same curve. A $10 increase on a $20 product is a 50 percent jump that users notice at once, while the same $10 on a $200 product is 5 percent and may pass unnoticed.
Why do our senses track proportions?
Senses have to cover enormous ranges. Human hearing handles sounds from a whisper to a jet engine, an intensity range of roughly a trillion to one. A system that responded to fixed differences could not work across that span, so it responds to ratios. This is why sound is measured in decibels and why astronomers have ranked star brightness on a logarithmic magnitude scale since the second century BCE.
How did Fechner turn this into a logarithm?
Gustav Fechner, who founded psychophysics with his 1860 book Elements of Psychophysics, treated each just noticeable difference as one equal step of sensation. Weber's law says each step needs a fixed proportion more stimulus. Adding up equal steps across proportional increases gives a logarithmic curve. In 1957, S. S. Stevens proposed a power law as an alternative that fit many senses better.
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