Representing Number Sets: Encoding Statistical Properties

Representing Number Sets: Encoding Statistical Properties Patrick F. Cravalho (pcravalh@kent.edu), Bradley J. Morris (bmorri20@kent.edu), Christopher A. Was (cwas@kent.edu) Department of Educational Psychology, 405 White Hall Kent, OH 44242 USA Amy M. Masnick (Amy.M.Masnick@hofstra.edu) Department of Psychology, Hauser Hall Hempstead, NY 11549 USA Abstract when asked to make number comparisons, people implicitly (i.e., without deliberate effort) represent the properties of number sets, such as means and variance, rather than information about individual numbers. Previous research suggests that people often recall individual items when sets are smaller than four and aggregate set features for sets larger than four. One intriguing possibility is that the process of aggregating sets creates summary representations that maintain the statistical properties of the set itself. For sets of numbers, this process might implicitly create approximate means. We report the results of two experiments investigating memory for number sets and its relation to working memory and metacognitive monitoring. In both experiments, participants were shown a series of data sets that varied in size (4, 6, or 8 numbers) and variance (10% or 20% of the mean) and were then presented with an actual value from the set and the set mean. In Experiment 1, participants were asked to select the actual value, and in Experiment 2, participants were asked to select the set mean. Results indicated that the proportion of correct selections and metacognitive confidence decreased with set size. Working memory was related to performance only when the set size was 6. The findings suggest that participants often erroneously reported the set mean as being a member of the set and that this effect increased for sets larger than four. The findings suggest that the process of aggregating number sets results in approximate means. Sets vs. Individuals A well-replicated finding is that people individuate items (i.e., represent individual values within a set) for sets smaller than four (Scholl, 2001). For example, given a set of three dots, participants are likely to remember the individual dots rather than the properties shared by the dots (Airely, 2001). Given sets larger than four, people aggregate items, or average over sets retaining information about set features (Scholl, 2001; Masnick & Morris, 2008). For example, given a set of a dozen dots of various sizes, participants were likely to erroneously recall a dot that represents the average size, rather than correctly recall an actual dot from the set they viewed (Airely, 2001). Earlier research suggested two separate memory stores for individual versus aggregated sets. For example, Feigenson, Dehaene, and Spelke (2004) suggested one system for individuating small values (< 4) and another system for aggregating larger sets. However, recent research (Hyde, 2011) suggests that the difference in representation may be functional (i.e., due to differences in strategy) rather than structural (i.e., separate stores). The process of aggregating appears to maintain information about the statistical properties of sets. Previous research on category learning provides evidence that aggregating over individual objects results in a prototype, or a most representative set member. Researchers have suggested that the process of aggregating yields “a measure of central tendency” (e.g., an average or prototypical category member; Medin, Altom, & Murphy, 1984, p. 334). Early research into prototypes suggested that when people aggregate across multiple exemplars, they extract a prototypical category member that represents an average of category features (Medin et al., 1984; Nosofsky, Denton, Zaki, Murphy-Knudsen, & Unverzagt, 2012). For example, a robin is often seen as a prototypical bird because it shares most of the features common among birds. Numbers provide an intriguing extension of this work because aggregation of sets may produce approximate Keywords: Number sets; Representation; Metacognition. Introduction Data interpretation (i.e., interpreting numbers in context) is critical to everyday reasoning. Numbers are represented in two ways, as an approximate magnitude and as a symbolic value (Dehaene, 2009). Previous research suggests that people compare single numbers as analog magnitudes, meaning that numbers are represented as approximate rather than exact values. Key evidence for this argument is that comparisons are faster and more accurate as the ratio of difference between values increases (Feigenson, Dehaene, & Spelke, 2004; Hyde, 2011). Yet little is known about how people represent numerical data sets. Previous research involving the cognitive processing of number sets suggests that people’s comparisons are highly related to the statistical properties of the sets. Specifically, as the ratio of mean differences between number sets increase and variance within each set decreases, comparisons to assess which set has the higher mean become faster and more accurate (Masnick & Morris, 2008). These findings suggest that