The Turtleback Diagram for Conditional Probability

Open Journal of Statistics, 2018

We elaborate on an alternative representation of conditional probability to the usual tree diagram. We term the representation “turtleback diagram” for its resemblance to the pattern on turtle shells.

Adopting the set theoretic view of events and the sample space, the turtleback diagram uses elements from Venn diagrams—set intersection, complement and partition—for conditioning, with the additional notion that the area of a set indicates probability whereas the ratio of areas for conditional probability.

Once parts of the diagram are drawn and properly labeled, the calculation of conditional probability involves only simple arithmetic on the area of relevant sets.

We discuss turtleback diagrams in relation to other visual representations of conditional probability, and detail several scenarios in which turtleback diagrams prove useful.

By the equivalence of recursive space partition and the tree, the turtleback diagram is seen to be equally expressive as the tree diagram for abstract concepts.

We also provide empirical data on the use of turtleback diagrams with undergraduate students in elementary statistics or probability courses.

We discussed our graphical tool in the context of other graphical models for conditional probability, and carried out case studies on over 200 students of elementary statistics or probability classes.

Our case study results are encouraging and the graph-based approaches could potentially lead to significant improvements in both the students’ understanding of conditional probability and problem solving.

Though the turtleback diagram appears very different from the tree diagram, we are able to unify them and show their equivalence in terms of semantics.

Our discussion suggests a simple framework for visualizing abstract concepts, that is, a suitable graph representation of the abstract concept followed by a simple post-processing in the visual-brain system.

A good visualization idea needs to balance both. We are able to use such a framework to interpret the difficulty encountered by the tree diagram, and aid our development of the turtleback diagram.