Developing aspects of distribution in response to a media-based statistical literacy task

1. Background Research into students' beginning intuitions about distribution has generally been associated with variation in single variable settings (e.g., Ben-Zvi & Sharett-Amir, 2005; Watson & Kelly, 2005) and has often focused on graphing attributes of students' created representations. Kelly and Watson (2002) for example found that students' graphs to represent the imagined outcomes of repeated sampling trials in a probability setting ranged from idiosyncratic drawings of the physical scenario, to time-series type graphs inconsistently justified by " more " of a certain characteristic, to informal graphing based on " middle, " to a conventional distribution recognizing variation and center. In sampling or measurement investigations focusing mainly on single variable distributions, Shaughnessy (2006) described six aspects of variation as (i) extremes or outliers, (ii) change over time, (iii) the whole range, (iv) the likely range, (v) distance or difference from some fixed point, or (vi) sums of residuals. These descriptors are not seen as hierarchical and inform this study to assist in characterizing the story that students attempt to tell with the graphs they produce to represent a verbal description of covariation. This paper, in moving from the consideration of the distribution associated with a single variable to that associated with two variables, builds on earlier research on correlational reasoning and its representation (e.g., Ross and Cousins, 1993a, 1993b). Shaughnessy's (2006) aspects of variation also apply when two variables are involved, seen as the trend in the relationship between the two variables, and seen as deviations from the trend. Various researchers asked students to create graphical distributions from data values. Brasell and Rowe (1993), for example, asked physics students to construct a graph of five paired values representing the heights from which a ball was dropped and the height to which it rebounded; they found students drew pictures, produced poorly labeled graphs, or plotted points, but rarely gave evidence of graphing to show a trend. Rather than starting with data values, Mevarech and Kramarsky (1997) asked grade 8 students to graph four different verbal claims about trend relationships of time spent studying and the marks received at school. Whereas slightly over half of students appropriately graphed a positive, negative, or no association, fewer than half graphed a curvilinear association. The most common errors observed related to graphing only a single point, to graphing only a single variable, or to graphing an increasing function regardless of the conditions set in the …