Confidence judgments during ratio comparisons reveal a Bayesian bias

Rational numbers are essential in mathematics and decision-making but humans often and erroneously rely on the magnitude of the numerator or denominator to determine the relative size of a quotient. The source of this flawed whole number strategy is poorly understood. Here we test the Bayesian hypothesis that the human bias toward large values in the numerator or denominator of a ratio estimate is the result of higher confidence in large samples. Larger values are considered a better (more certain) instance of that ratio than the same ratio composed of smaller values. We collected confidence measures explicitly (Experiment 1) and implicitly (Experiment 2) during subjects' comparisons of non-symbolic proportions (images with arrays of orange and blue dots). We manipulated the discernibility of the fractions to control difficulty and varied the cardinality and congruency of the numerators, denominators, and ratio values (e.g. 8/20 vs. 5/10 and 16/40 vs. 10/20). The results revealed that subjects' confidence during ratio comparisons was modulated by the numerical magnitude of the fraction's components, consistent with a Bayesian perception of relative ratios. The results suggest that the large number bias could arise from greater confidence in large samples.