Ensemble averaging of symbolic numbers: Criterion mean judgments beat heuristics

Can observers compute an ensemble mean from symbolic numbers, or is averaging-like performance driven by heuristic shortcuts? We addressed this question using a criterion-based design that yields clear predictions for mean-based integration. In two experiments, participants viewed a $2\times 2$ array of four Arabic digits and decided, while timed, whether the arithmetic mean of the set was smaller or larger than 5. In Experiment 1 ( $N=40$ ), accuracy and response time showed a robust ensemble-distance effect: decisions were slowest and least accurate when the set mean was closest to 5 and improved monotonically with mean distance. Item-level model comparisons further indicated that mean distance explained behavior better than prominent heuristic alternatives based on the most extreme digit or the majority of digits above versus below the criterion, and within-set dispersion produced an additional, independent impairment. Experiment 2 ( $N=40$ ) provided a stronger diagnostic test by constructing stimulus regimes that rendered heuristic cues uninformative (majority ties) or misleading (extreme-item conflict) while matching mean distance and dispersion across regimes. Across regimes, model comparisons again favored mean distance for both accuracy and response time. Critically, the distance effect persisted when majority information carried no diagnostic value and when the most extreme digit favored the incorrect response. These results support genuine mean-based integration of symbolic magnitudes, while showing that heuristic strategies have only a secondary role apparent only under increased uncertainty. OPEN PRACTICE STATEMENT: Data and stimuli can be found here https://osf.io/xnq92 , and none of the experiments was preregistered.