Working with Circular Data: A Tutorial for Cognitive and Behavioral Research

Introductory statistics courses and conventional methods in behavioral and cognitive science are typically restricted to Euclidean space without explicitly stating this, leaving many researchers unaware of how to handle data with circular or periodic properties (e.g., time of day or orientation on a screen). This tutorial provides an accessible yet rigorous introduction, designed to bridge the gap between statistical theory and practical application. After discussing different ways of representing circular data mathematically and fundamental principles for working with them, we explain angular and vector representations and operations, including key considerations for implementing them in computer code. We then survey methods for visualizing circular data, including linear and polar plots and presentation of summary statistics. Next, we introduce circular measures of central tendency, dispersion, and shape: how they relate to their Euclidean counterparts, and why they differ. Finally, we turn to more advanced topics of the kind needed to construct models of circular data. We introduce the most common circular distribution families, including circular analogs of the Gaussian distribution and their properties, as well as flexible distribution families that can describe skewed and heavy-tailed data. We survey common applications in the supplementary materials. The tutorial takes a method-oriented approach, not tied to specific software, but highlighting factors relevant to experimental design, data analysis, and statistical modeling. A consistent theme is the need to rigorously account for the periodicity of circular data. We also provide a consistent notation for circular statistics and parameters, addressing the lack of established conventions in the field.