Calculation Efficiencies for Mean Numerosity

Phenomenology, Quantity, and Numerosity

There are many situations in which we interact with collections of objects, from a crowd of people to a bowl of blackberries. There is an experience of the quantity of these items, although not a precise number, and we have this impression quickly and effortlessly. It can be described as an expressive property of the whole. In the literature, the study of this sense of numerosity has a long history, which is reviewed here with examples. I argue that numerosity is a direct perceptual experience, and that all experiences of numerosity, not only estimations, are affected by perceptual organisation.

Spatial number estimation has a higher linear range than temporal number estimation; differential affordances for subdivision might help to explain why

Estimation of visuospatial number typically has a limited linear range that goes well beyond the subitizing range but typically not beyond 20 items without calibration procedures. Three experiments involving a total of 104 undergraduate students, each tested once, sought to determine if the limit on the linear range represented a capacity limitation of a linear accumulator or might be the result of a strategy based on subdividing spatial displays into potentially subitizable subsets. For visual and auditory temporal numbers for a large range of numbers (2–58; Experiment 1), the (unbiased) linear range was found to be quite restricted (three or four items). Using matched linear spatial number stimuli (Experiment 2), the linear range observed extended to about nine or 10 items. Experiment 3 compared estimates when simultaneous two-dimensional spatial number displays were presented briefly, with estimates for identical displays that accumulated over time. The linear range of estimates for accumulating spatial displays reached only 11 items, whereas that for briefly presented displays extended to about 20 items. These results suggest that the limit on the linear range is not simply a capacity limitation in a linear accumulator. Rather, they support the idea that linear spatial number estimation for the range from five to 20 may be based on subdividing the display into a subitizable number of (potentially) subitizable groups, even if those groups are not outwardly marked.

Greater sensitivity in separation discrimination with closer spacing of separation levels tested

We explored the effects of spacing in the levels of separation tested in a separation discrimination task. Participants indicated, for pairs of test circles, whether the separation between them was greater than a standard separation. A critical set of equally-spaced separation levels was tested in two conditions. In one condition additional separation levels were interleaved between the critical levels. In the other condition additional separation levels were not interleaved. Overall, the same average level and range of levels were tested in the two conditions, and the levels tested were equally spaced in both conditions. Critically, the levels tested were more closely spaced in one condition than the other. The sensitivity of the discrimination was greater in the condition with the more as opposed to less-closely spaced levels of separation. We suggest an explanation under which separation is assessed from the number of "separation fields" between the points at which the test stimuli register and under which the separation fields are smaller or more densely distributed when the levels of separation tested are more as opposed to less-closely spaced.

Multiple concurrent centroid judgments imply multiple within-group salience maps

Subjects viewed a brief flash of 8-24 dots of either two or three colors randomly arrayed. Their task was to move a mouse cursor to the centroid (center-of-gravity) of each color in a pre-designated order. Conventional and idea-detector analyses show that subjects accurately judged all three centroids utilizing an astounding 13/24 stimulus dots, with only a modest loss of accuracy compared to judging a single-predesignated color centroid. The ability to concurrently compute three centroids is important because it is believed that centroid judgments are made on salience maps that record only salience and are ignorant of the features that produced the salience. Our explanation, instantiated in a computational model of salience processing, is that subjects have three salience maps. Dots are initially segregated into three groups according to color, then each color-group is recorded on a different salience map to compute a centroid. In Part 2, the data are analyzed in terms of Attention Operating Characteristics to characterize impairments in subjects' color-attention filters (mostly insignificant) and encoding efficiency (20% drop for the hardest task) in making multiple versus single centroid judgments. A new, more sensitive analysis measured five sources of subject error variance, four independent, additive sources of error variance: imperfect color-attention filters; a Bayesian-like bias towards a central tendency; storage, retrieval, and cursor misplacement error; a large residual error due mostly to inefficient encoding; and fifth, an interactive source - error in all four components that increases when multiple centroid judgments versus a single centroid judgment are required on each trial. SIGNIFICANCE STATEMENT: An important brain process is a salience map, a representation of the relative importance (salience) of the locations of visual space. It is needed to guide where to look next, for computing the center (technically "centroid") of a cluster of items, and for other important computations. Here we show that in a brief flash of dots of three different colors, randomly interleaved, subjects can compute all three centroids. As a single salience map cannot discriminate dots of different colors, accurately reporting three centroids demonstrates that subjects have not just one, as is commonly believed, but at least three salience maps.

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.

Five dichotomies in the psychophysics of ensemble perception

Whereas psychophysicists may formulate hypotheses about appearance, they can only measure performance. Bias and imprecision in psychophysical data need not necessarily reflect bias and imprecision in perception. Sensory systems may exaggerate the differences between each item and its neighbors in an ensemble. Alternatively, sensory systems may homogenize the ensemble, thereby removing any apparent differences between neighboring items. Ensemble perception may be involuntary when observers attempt to report the identities of individual items. Conversely, when asked to make a (voluntary) decision about the ensemble as a whole, observers may find it very difficult to compute statistics that are based on more than a very small number of individual items. Modeling decisions about prothetic continua, such as size and contrast, can be tricky because sensory signals may be distorted before and/or after voluntarily computing ensemble statistics. With metathetic continua, such as spatial orientation, distortion is less problematic; physically vertical things necessarily appear close to vertical and physically horizontal things necessarily appear close to horizontal. Decision processes are corrupted by noise that, like distortion, may be added to sensory signals prior to and/or after voluntarily computing ensemble statistics.