1  Introduction

Uncertainty visualisation is a relatively new field, but it has been poorly defined. Uncertainty itself is a vague term; each paper seems to take the meaning of it as a given, so the visualisation of it is fraught with disagreement and conflict over definitions, applications, and best practices. Incorporating uncertainty into our visualisations is important for transparent graphics and improved decision making, with some authors considering ignoring uncertainty to be tantamount to fraud (Hullman 2020). Despite the importance of uncertainty visualisation, the field has struggled to create an overarching cohesive theory, with current best practices in plot design remaining ad hoc and context specific (MacEachren et al. 2005). These issues would be resolved if uncertainty were formalised using existing structures for statistical graphics, such as the grammar of graphics (Wilkinson 2005) and its implementation in ggplot2 (Wickham 2010). The formalisation of uncertainty in the grammar of graphics has remained difficult due to the confusing nature of uncertainty, a term which has as many definitions as there are discussions on the topic (Spiegelhalter 2017).

This thesis attempts to close these gaps by providing a comprehensive analysis of uncertainty visualisation, including its definition, practical application, and evaluation. First, we deal with the vague definition of uncertainty and interrogate the current meaning of the term “uncertainty visualisation”. Building on foundations in statistical inference and graphics, we redefine what it means to visualise uncertainty and establish a statistical foundation for the goals of the field. Next, we leverage this foundation to formalise uncertainty visualisation in the grammar of graphics, implemented in the ggplot2 extension ggdibbler. We show that the formalisation goes beyond mathematical pedantry as it facilitates exploratory data analysis of uncertain data, an application that has remained just out of reach of the field for decades. Finally, we tie these ideas together in an evaluation study on the perception of uncertainty visualisations, and show that the formalisation we designed is not only flexible, but provides a framework for testing and validating these graphics. This validation is done using a novel experiment design, by drawing parallels in the goals of uncertainty visualisation and standard colour blind tests, repurposing the standard Ishihara colour blind test for uncertainty visualisation. Statistical graphics exist at the intersection of philosophy, statistics, computer science, and psychometrics, and a thorough dive into the problems facing uncertainty visualisation will require us to touch on all four fields.

1.1 Thesis Outline

The thesis is structured as follows.

Chapter 2 provides a comprehensive review of the uncertainty visualisation literature. Existing reviews in the field take the purpose of an uncertainty visualisation to be self-evident, which results in a large amount of conflicting information. The most common approach communicates uncertainty as a probability or a distribution, and focuses on showing the uncertainty as an isolated variable disconnected from its context. We contrast these visualisations with approaches that view uncertainty as noise that should be incorporated to give a holistic view of our data. We coin the term “signal suppression” to describe a visualisation that is designed for preventing false conclusions, as the approach demands that the signal (i.e., the conclusions drawn from the estimates) is suppressed by the noise (i.e., the variance on those estimates). We provide motivation for viewing signal suppression as the most worthwhile goal in uncertainty visualisation, and argue that visualisations that display uncertainty as an isolated variable should not be considered uncertainty visualisations at all. We further discuss difficulties in creating and evaluating the effectiveness of these plots, which further motivates the following chapters of this thesis.

Chapter 3 introduces a mathematical framework for uncertainty visualisation, as well as a new R package, ggdibbler. In Chapter 2 we discussed the run-on effects of the ambiguous definition of uncertainty. These problems included difficulty incorporating uncertainty into the grammar of graphics framework, which leads to software that does not always behave in ways that users would expect. Chapter 3 closes this gap by discussing the theoretical framework required to integrate uncertainty into the grammar of graphics, along with the desirable statistical properties that go along with this framework. This framework is demonstrated in ggdibbler, a ggplot2 extension that allows for flexible uncertainty visualisation for exploratory data analysis. The software allows users to replace any vector of variables in a ggplot2 plot with a vector of random variables, and get an uncertain version of a plot. These plots include uncertainty in such a way that they align with the goals suggested by Chapter 2, and reduce the visibility of statistically invalid signals.

Chapter 4 presents a user study that evaluates human perception of uncertainty visualisations. Chapter 2 highlighted the difficulty in evaluating uncertainty visualisations, as the uncertainty must be evaluated as noise rather than signal. Correctly evaluating uncertainty visualisations, therefore, requires us to design an experiment that measures uncertainty as a latent variable. This section applies the suggested experimental methods and identifies which graphics introduced in Chapter 3 are effective tools for signal suppression. This experiment confirms many of the hypothesis proposed by the literature review in Chapter 2, and suggests that the visualisations made by Chapter 3 are statistically valid.

The material in Chapter 2 was submitted to Annual Reviews of Statistics and Its Applications as an invited contribution to Volume 14 of the Journal. The ggdibbler software outlined in Chapter 3 is published on CRAN. The contribution in Chapter 3 of this thesis was presented at useR! 2025 in Durham, North Carolina, in August 2025 and ASC 2026 in Perth, Western Australia, in December 2025. The material in Chapter 3 is prepared for submission to the Journal of Computational Graphics and Statistics. The material in Chapter 4 has been submitted to IEEE VIS 2026.

Hullman, J. (2020), “Why authors don’t visualize uncertainty,” IEEE Transactions on Visualization and Computer Graphics, IEEE Computer Society, 26, 130–139. https://doi.org/10.1109/TVCG.2019.2934287.

MacEachren, A. M., Robinson, A., Hopper, S., Gardner, S., Murray, R., Gahegan, M., and Hetzler, E. (2005), “Visualizing geospatial information uncertainty: What we know and what we need to know,” Cartography and Geographic Information Science, 32, 139–160. https://doi.org/10.1559/1523040054738936.

Spiegelhalter, D. (2017), “Risk and uncertainty communication,” Annual Review of Statistics and Its Application, 4, 31–60. https://doi.org/10.1146/annurev-statistics-010814-020148.

Wickham, H. (2010), “A layered grammar of graphics,” Journal of Computational and Graphical Statistics, 19, 3–28. https://doi.org/10.1198/jcgs.2009.07098.

Wilkinson, L. (2005), The grammar of graphics, Berlin, Heidelberg: Springer-Verlag. https://doi.org/10.1007/0-387-28695-0.