Bipolar Distribution


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A bipolar distribution is a specific type of bimodal distribution that has two distinct peaks offering contradictory information. In some cases, the term may also refer to the Watson bipolar distribution.

Bimodal vs. Bipolar Distribution

A bimodal distribution is a distribution with two distinct peaks or modes. The modes can be anywhere on the distribution.

The above image shows that the bulk of students get either 3 or 8 answers right, with a variety of other possibilities. The graph provides a little information (but not much) about the quality of the test or student performance.

In contrast, the following image shows a bipolar distribution of movie ratings where movie goers either hate or love the movie [1]. As such, it offers a wealth of contradictory information; while we know everyone hated or loved the movie, we cannot make any determination at all about the quality of the movie.

This distribution can also be described as an extreme U-shaped distribution [3].

Watson bipolar distribution

The Watson bipolar distribution  (or simply the Watson distribution) is a single-direction distribution used in directional data analysis, where bipolar or girdle distribution are required [2]; Girdle distributions are measurement bands on the stereographic projection, scattered on the sided of a great circle [4].

References

[1] STAT 22000 Lecture Slides Variability in Estimates & Central Limit Theorem. http://www.stat.uchicago.edu/~yibi/teaching/stat220/17aut/Lectures/L14.pdf

[2] Mardia, K. V., & Dryden, I. L. (1999). The Complex Watson Distribution and Shape Analysis. Journal of the Royal Statistical Society. Series B (Statistical Methodology)61(4), 913–926. http://www.jstor.org/stable/2680713

[3] Klein, Ingo, 2017. “(Generalized) maximum cumulative direct, paired, and residual Φ entropy principle,” FAU Discussion Papers in Economics 25/2017, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics

[4] Waldron, J. (2020). Data in Structural Geology. https://courses.eas.ualberta.ca/eas421/lecturepages/orientation.html