Efficiencies for the statistics of size discrimination.

Different laboratories have achieved a consensus regarding how well human observers can estimate the average orientation in a set of N objects. Such estimates are not only limited by visual noise, which perturbs the visual signal of each object's orientation, they are also inefficient: Observers effectively use only √N objects in their estimates (e.g., S. C. Dakin, 2001; J. A. Solomon, 2010). More controversial is the efficiency with which observers can estimate the average size in an array of circles (e.g., D. Ariely, 2001, 2008; S. C. Chong, S. J. Joo, T.-A. Emmanouil, & A. Treisman, 2008; K. Myczek & D. J. Simons, 2008). Of course, there are some important differences between orientation and size; nonetheless, it seemed sensible to compare the two types of estimate against the same ideal observer. Indeed, quantitative evaluation of statistical efficiency requires this sort of comparison (R. A. Fisher, 1925). Our first step was to measure the noise that limits size estimates when only two circles are compared. Our results (Weber fractions between 0.07 and 0.14 were necessary for 84% correct 2AFC performance) are consistent with the visual system adding the same amount of Gaussian noise to all logarithmically transduced circle diameters. We exaggerated this visual noise by randomly varying the diameters in (uncrowded) arrays of 1, 2, 4, and 8 circles and measured its effect on discrimination between mean sizes. Efficiencies inferred from all four observers significantly exceed 25% and, in two cases, approach 100%. More consistent are our measurements of just-noticeable differences in size variance. These latter results suggest between 62 and 75% efficiency for variance discriminations. Although our observers were no more efficient comparing size variances than they were at comparing mean sizes, they were significantly more precise. In other words, our results contain evidence for a non-negligible source of late noise that limits mean discriminations but not variance discriminations.