Random variables are the bread and butter of statistics, and visualisations are one of the most versatile tools in the field, so it is a wonder why we do not have a methodology for visualising random variables. This gap is particularly evident for exploratory data analysis (EDA). We address this gap by designing a mathematical framework for visualisation, which argues that we should consider visualisations to be continuous functions. In the case of random variable inputs, this means the visualisations should obey the continuous mapping theorem (CMT). By breaking the visual function down into its components, we are able to identify which parts of the mapping are ill-defined for random variable inputs and redefine them in a way that guarantees both the flexibility required for EDA and the statistical sensibility of CMT. This formalisation represents a complete integration of uncertainty into the grammar of graphics, which we show by implementing the theory in the R package, "ggdibbler". The ggdibbler software is a "ggplot2" extension that allows users to replace the data of any plotting function with a random variable, with the guarantee that the visualisation will have the same convergence properties as its underlying data.

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