Congratulations to Adam Cunningham at University at Buffalo for winning the Best Contributed Paper Award in the Statistics and Data Science Education Section at JSM 2023! His paper, “Probability Playground: Exploring Probability Distributions Through Interaction'' discusses the backstory, features, and design considerations for an interactive website that allows users to explore probability distributions and the relationships between them. Check the tool out here and the paper providing more details here. You can also learn more about Adam here. In this blog post we’ll cover some highlights.
Motivation
Adam was in a Biostatistics Masters program at the University at Buffalo learning about a variety of probability distributions, deriving them, working with them in a mathematical way, etc. but he had many questions about what they can look like in different scenarios and about the relationship between them. He had seen one-off online tools that let you play around with various distributions, tweaking a parameter here and there to see how it affected the shape of the distribution, and static maps of relationships between distributions, but he wanted something that would enable further exploration in a more unified way.
Features
Enter the Probability Playground. The “Map” page of the tool shows each distribution and the connections between them. The distributions are color coded by type of distribution, and the links between distributions are also color coded to describe types of transformations or relationships. Each distribution and each link between distributions is clickable to take you to a new page with more detail.
Let’s take the lognormal distribution for instance. The first panel gives an overall summary of the distribution and its parameters. It also provides some examples that when clicked, activate the interactive plots of the distribution in the second and third pane. This gives you a place to start exploring different instances of the distribution to build some intuition about its shape.
In the second panel you can explore the PDF and CDF of the distribution as well as watch a simulation of the variable in real time. The visual display of the data-generating process is consistent across distributions, so the user can build intuition across distributions as well as within a single one.
The right-most panel shows various relationships that are related to this distribution, with a dropdown so that you can explore more than one. For example, you can work through the intuition of the product of independent lognormal random variables leading to yet another lognormal random variable in this pane.
There are over 100 proofs about properties of the distributions and of the relationships between them baked into the tool as well.
Design Considerations
A unique feature of this interactive tool is that a user can hold the mean constant while changing the variance and hold the variance constant while changing the mean. This gives a lot of flexibility in exploration to understand the relationship between the distribution’s parameters, its center, and its spread. While a user is playing around with the distributions, there is automatic scaling of the axis so that the space of the distribution is appropriately displayed as it changes. The logistics of this are non-trivial!
Adam wanted his tool to be accessible to users who use a keyboard for moving about on a web page rather than relying on the finer motor control needed to use a mouse. Each interactive element that a user can play with to explore the distributions has a keyboard navigation equivalent.
We’ll let you take it from here and further explore. If you end up using this tool in your classroom or as a learner we’d love to hear about it (and we’re sure Adam would too!).