Benefits of Graphical and Symbolic Representations for Learning and Transfer of Statistical Concepts

Benefits of Graphical and Symbolic Representations for Learning and Transfer of Statistical Concepts David W. Braithwaite (dwbraith@indiana.edu) Robert L. Goldstone (rgoldsto@indiana.edu) Department of Psychological and Brain Sciences, 1101 E. 10th Street Bloomington, IN 47405 USA Abstract Past research suggests that spatial configurations play an im- portant role in graph comprehension. The present study in- vestigates consequences of this fact for the relative utility of graphs and tables for interpreting data. Participants judged presence or absence of various statistical effects in simulated datasets presented in various formats. For the statistical ef- fects introduced earlier in the study, performance was better with graphs than with tables, while for the effect introduced last in the study, this trend reversed. Additionally, in the later sections of the study, responses with graphs, but not tables, reflected increasing influence from the presence of stimulus features which had been relevant earlier in the study, but were no longer relevant. The findings suggest that graphs, relative to tables, may better facilitate perception of complex relation- ships among data points, but may also bias readers more strongly to favor some perspectives over others when inter- preting data. Keywords: representations; graphs; tables; mathematics; sta- tistics; human factors Introduction Humans have devised a variety of different formats for ex- ternally representing information. Often, the same infor- mation may be represented in multiple representations that are informationally equivalent, in that each may be recon- structed perfectly on the basis of any other. Despite such equivalence, different representations may support perfor- mance of specific cognitive tasks at different levels of effi- ciency. Such differences have important implications for the selection and design of external representations. The present study explores such differences with respect to graphs and tables, two of the most commonly-employed representational formats for quantitative information in a variety of fields. The relative advantages of graphs and tables have been the subject of extensive research. Tables appear to be at least as effective as graphs with respect to point reading tasks, which require one to estimate or read off individual data points (Meyer, Shamo, & Gopher, 1999; Porat, Oron-Gilad, & Meyer, 2009; Vessey & Galletta, 1991). However, graphs have often shown advantages for tasks involving complex relationships between multiple data points, such as estimating or comparing differences between points (Schonlau & Peters, 2012; Vessey & Galletta, 1991), projecting trends (Meyer et al., 1999), and detecting changes in function parameters (Porat et al., 2009). Models of graph comprehension (Carpenter & Shah, 1998; Ratwani, Trafton, & Boehm-Davis, 2008) suggest a possible explanation for the latter findings. According to these models, spatial configurations of data points are the raw material on which graph comprehension processes op- erate. Importantly, some configurations may be directly perceived as basic visual features (Pomerantz & Portillo, 2012), allowing relationships between points to be “read off” directly without first encoding each point separately (Carpenter & Shah, 1998; Porat et al., 2009). For example, distances between points may be used to determine or esti- mate differences in the values they represent, without the need to encode those individual values at all (Pinker, 1990). Thus, in graphs, spatial configurations can act as cues for recognizing relationships between data points. Because such cues are unavailable or less salient in tables, this prop- erty of graphs can account for their observed advantages in conveying relationships among data points. Many studies comparing task performance with graphs and tables have employed univariate datasets (Meyer et al., 1999; Porat et al., 2009). Consideration of bivariate data introduces another difference between graphs and tables. In graphs of bivariate data, there is a representational asym- metry between the two independent variables, in that one is often laid out along a spatial axis, typically the x-axis, while the other is typically represented by a non-spatial visual feature such as line color or thickness. For tables, on the other hand, such representational asymmetry is reduced, because the levels of both independent variables are laid out along spatial axes, albeit horizontal in one case and vertical in the other. Can such representational asymmetries as exist in graphs of bivariate data lead to performance asymmetries in tasks involving one or the other variable? A few studies have provided evidence in the affirmative (Carpenter & Shah, 1998; Shah & Freedman, 2011). For example, Shah and Freedman (2011) found that when asked to interpret graphs of bivariate data, participants were more likely to describe main effects of the variable depicted in the legend than of that depicted on the x-axis, and were more likely to describe interaction effects as moderating effects of the legend varia- ble on the effect of the x-axis variable than vice versa. Such representational asymmetries in graphs, together with the intuition that these asymmetries are reduced in ta- bles, suggest that performance asymmetries between tasks relating to one or the other independent variable in bivariate data should be greater for graphs than for tables. While a few studies have compared performance with graphs and tables on tasks involving bivariate data (Schonlau & Peters, 2012; Vessey & Galletta, 1991), the specific issue of how