Repulsive Serial Effects in Visual Numerosity Judgments

Wave-like patterns in parameter space interpreted as evidence for macroscopic effects resulting from quantum or quantum-like processes in the brain

Data from eight numerosity estimation experiments reliably exhibit wave-like patterns in plots of the standard deviations of the response times along the abstract parameter of the magnitude of the error in the numerosity estimation. An explanation for this phenomenon is proposed in terms of an analogy between response times and error magnitude on one hand, and energy and position of quantum particles on the other, constructed using an argument for an overlap between the mathematical apparatus describing Hopfield-type neural networks and quantum systems, established by some researchers. Alternative explanations are presented within the traditional explanatory framework of oscillations due to neural firing, involving hypothetical mechanisms for converting oscillation patterns in time to oscillation patterns in the space of an abstract parameter, such as the magnitude of the error during numerosity estimation. The viability of the proposal of causal influences propagating from the microscale of quantum phenomena to the macroscale of human behavior, needed for the first type of explanation, is exemplified by the phenomenon of magnetoreception in some species of birds, which is allegedly quantum in nature.

Enumerating the forest before the trees: The time courses of estimation-based and individuation-based numerical processing

Ensemble perception refers to the ability to report attributes of a group of objects, rather than focusing on only one or a few individuals. An everyday example of ensemble perception is the ability to estimate the numerosity of a large number of items. The time course of ensemble processing, including that of numerical estimation, remains a matter of debate, with some studies arguing for rapid, "preattentive" processing and other studies suggesting that ensemble perception improves with longer presentation durations. We used a forward-simultaneous masking procedure that effectively controls stimulus durations to directly measure the temporal dynamics of ensemble estimation and compared it with more precise enumeration of individual objects. Our main finding was that object individuation within the subitizing range (one to four items) took about 100-150 ms to reach its typical capacity limits, whereas estimation (six or more items) showed a temporal resolution of 50 ms or less. Estimation accuracy did not improve over time. Instead, there was an increasing tendency, with longer effective durations, to underestimate the number of targets for larger set sizes (11-35 items). Overall, the time course of enumeration for one or a few single items was dramatically different from that of estimating numerosity of six or more items. These results are consistent with the idea that the temporal resolution of ensemble processing may be as rapid as, or even faster than, individuation of individual items, and support a basic distinction between the mechanisms underlying exact enumeration of small sets (one to four items) from estimation.

Synergy between research on ensemble perception, data visualization, and statistics education: A tutorial review

In the age of big data, we are constantly inventing new data visualizations to consolidate massive amounts of numerical information into smaller and more digestible visual formats. These data visualizations use various visual features to convey quantitative information, such as spatial position in scatter plots, color saturation in heat maps, and area in dot maps. These data visualizations are typically composed of ensembles, or groups of related objects, that together convey information about a data set. Ensemble perception, or one's ability to perceive summary statistics from an ensemble, such as the mean, has been used as a foundation for understanding and explaining the effectiveness of certain data visualizations. However, research in data visualization has revealed some perceptual biases and conceptual difficulties people face when trying to utilize the information in these graphs. In this tutorial review, we will provide a broad overview of research conducted in ensemble perception, discuss how principles of ensemble encoding have been applied to the research in data visualization, and showcase the barriers graphs can pose to learning statistical concepts, using histograms as a specific example. The goal of this tutorial review is to highlight possible connections between three areas of research-ensemble perception, data visualization, and statistics education-and to encourage research in the practical applications of ensemble perception in solving real-world problems in statistics education.