The reason a percentage feels different to a fraction

A safety notice states that twenty per cent of users experience a delay. A second version states that one in five users experience a delay. Readers of the second version rate the problem as more serious and are more likely to act on it. The arithmetic is identical and the perceived magnitude is not.

What everyone sees

Both statements are treated as interchangeable, and in a strict mathematical sense they are. The assumption is that a well-informed reader converts one to the other automatically, so the choice of format is a matter of house style rather than of substance.

What is actually happening

Slovic, Monahan and MacGregor found that clinicians judged a patient more dangerous when risk was expressed as twenty out of a hundred rather than twenty per cent, and attributed the gap to the imagery that frequency formats evoke. Gigerenzer and Hoffrage showed that natural frequencies improve accuracy in Bayesian problems where percentages produce errors, because frequencies preserve the reference class. A percentage is a normalised ratio with the population removed; a frequency retains a countable group, and a countable group can be pictured while a ratio cannot.

Why it stays hidden

The asymmetry hides because the conversion is trivial. Any reader can turn one in five into twenty per cent in a second, so it seems impossible that the formats could carry different weight. But the conversion happens after the impression has formed, and the impression is produced by whichever format was presented. Knowing the equivalence does not undo the imagery that arrived first.