Approximate number perception is noisy, but it is unclear what kind of underlying process the noise reflects. Here we provide evidence that approximate number estimation should be thought of as a sampling procedure. We show that the the average of two approximate number estimates of the same stimulus tends to outperform either estimate alone; additionally, the average difference between the two estimates of a given number linearly increases as a function of number, consistent with Weber’s law. Finally, we provide evidence that people report confidence ranges consistent with Weber’s law. This suggests that they represent a distribution of possible responses even on a single trial.

Abstract: Humans are both the scientists who discover psychological laws and the thinkers who behave according to those laws. Oftentimes, when our natural behavior is in accord with those laws, this dual role serves us well: our intuitions about our own behavior can serve to inform our discovery of new laws. But, in cases where the laws that we discover through science do not agree with the intuitions and biases we carry into the lab, we may find it harder to believe in and adopt those laws. Here, we explore one such case. Since the founding of psychophysics, the notion of a Just Noticeable Difference (JND) in perceptual discrimination has been ubiquitous in experimental psychology—even in spite of theoretical advances since the 1950’s that argue that there can be no such thing as a threshold in perceiving difference. We find that both novices and psychologically educated students alike misunderstand the JND to mean that, below a certain threshold, humans will be unable to tell which of two quantities is greater (e.g., that humans will be completely at chance when trying to judge which is heavier, a bag with 3000 grains of sand or 3001). This belief in chance performance below a threshold is inconsistent with psychophysical law. We argue that belief in a JND is part of our intuitive theory of psychology and is therefore very difficult to dispel.