Abstract
In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2 × 2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch structures, these two visualizations (in contrast to the 2 × 2 table) can even simultaneously display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2 × 2 tables and (double-)trees have a decisive disadvantage: While joint probabilities [e.g., P(A∩B)] are represented in 2 × 2 tables but no conditional probabilities [e.g., P(A|B)], it is exactly the other way around with (double-)trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2 × 2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2 × 2 table, double-tree, or net diagram with absolute frequencies) are consistently superior to probability visualizations, and the frequency net performs as well as the frequency double-tree. Only the 2 × 2 table with frequencies—the one visualization that participants were already familiar with—led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, but no uniform format effect becomes visible. Here, participants reached the highest performance in the versions with probability 2 × 2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability.
Highlights
In this article the frequency net is presented as a new tool for simultaneously visualizing probabilities and frequencies, a capability that is not possible with the use of existing visualizations such as the 2 × 2 table, the tree diagram, and the double-tree
No visualization had the capacity to represent all 16 probabilities that can occur in a situation with two dichotomous characteristics and all frequencies simultaneously
The fact that the frequency net can enable visualization of (1) probabilities and frequencies and (2) joint probabilities and conditional probabilities is a didactic advantage because performing demanding additional calculations based on a net diagram is no longer necessary
Summary
Experimental cognitive psychology research on the effects of natural frequencies and visualizations focuses primarily on conditional probabilities, especially on Bayesian tasks like the famous mammography problem and similar cognitive illusions like the Monty Hall problem (Kahneman et al, 1982; Gigerenzer and Hoffrage, 1995; Goodie and Fantino, 1996; Hoffrage et al, 2000; Krauss and Wang, 2003; Barbey and Sloman, 2007; Spiegelhalter et al, 2011; Baratgin, 2015; Operskalski and Barbey, 2016; McDowell and Jacobs, 2017).conditional probability tasks, and especially Bayesian tasks are only one aspect of teaching probability at secondary schools and university. In the teaching of statistics at secondary school and university level, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2 × 2 tables and tree diagrams. Both visualizations can be depicted with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more than probability visualizations in Bayesian reasoning problems (Binder et al, 2015, 2018) Tree diagrams and their extensions to double-trees can even display both information formats simultaneously, which is an advantage from a pedagogical point of view
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