As the level of vehicle automation increases, drivers are more likely to engage in non-driving related tasks which take their hands, eyes, and/or mind away from the driving task. Consequently, there has been increased interest in creating Driver Monitoring Systems (DMS) that are valid and reliable for detecting elements of driver state. Workload is one element of driver state that has remained elusive within the literature. Whilst there has been promising work in estimating mental workload using gaze-based metrics, the literature has placed too much emphasis on point estimate differences. Whilst these are useful for establishing whether effects exist, they ignore the inherent variability within individuals and between different drivers. The current work builds on this by using a Bayesian distributional modelling approach to quantify the within and between participants variability in Information Theoretical gaze metrics. Drivers (N = 41) undertook two experimental drives in hands-off Level 2 automation with their hands and feet away from operational controls. During both drives, their priority was to monitor the road before a critical takeover. During one drive participants had to complete a secondary cognitive task (2-back) during the hands-off Level 2 automation. Changes in Stationary Gaze Entropy and Gaze Transition Entropy were assessed for conditions with and without the 2-back to investigate whether consistent differences between workload conditions could be found across the sample. Stationary Gaze Entropy proved a reliable indicator of mental workload; 92% of the population were predicted to show a decrease when completing 2-back during hands-off Level 2 automated driving. Conversely, Gaze Transition Entropy showed substantial heterogeneity; only 66% of the population were predicted to have similar decreases. Furthermore, age was a strong predictor of the heterogeneity of the average causal effect that high mental workload had on eye movements. These results indicate that, whilst certain elements of Information Theoretic metrics can be used to estimate mental workload by DMS, future research needs to focus on the heterogeneity of these processes. Understanding this heterogeneity has important implications toward the design of future DMS and thus the safety of drivers using automated vehicle functions. It must be ensured that metrics used to detect mental workload are valid (accurately detecting a particular driver state) as well as reliable (consistently detecting this driver state across a population).