Using Mathematica Demonstrations to Visualize Statistical Concepts

Statistics involves both calculations and conceptual understanding.  It’s common to use technology to help with the calculations.  This article is about using technology to help students understand the underlying concepts.  Finding ways to illustrate the ideas behind confidence intervals or hypothesis testing is difficult.  Even illustrating the roles of the mean and standard deviation can be a challenge.  My favorite tool for addressing these theoretical ideas is the Wolfram Demonstrations Project.

Wolfram Research, Inc’s best-known product is MathematicaMathematica is a very powerful symbolic algebra system.  Its graphical abilities provide a wonderful opportunity to illustrate mathematical concepts.  You could purchase a copy of Mathematica and invest hours learning to use Mathematica and developing illustrations to use in your classes.  Or you could visit www.demonstrations.wolfram.com and access the illustrations for free.  A Demonstration is a Mathematica program that uses Mathematica’s manipulate command.  The web site hosts over 7,000 Demonstrations illustrating every topic imaginable, including statistical topics.  To use the Demonstrations, you do have to download a plug-in for your browser.  Just click on the sign on the Demonstration you’re planning to run.  Once you download the plug-in, you can run the Demonstrations right from your browser.

I find useful Demonstrations by entering keywords in the search box.  Warning: The keyword “statistics” yields over 450 Demonstrations and “probability” yields over 500. Most of these are not appropriate for my course. I find more specific searches (like “Normal distribution” or “confidence interval”) much more useful.  Some of my favorite statistical Demonstrations are: Influential Points in RegressionThe Normal Distribution, How Do Confidence Intervals Work?, and Hypothesis Tests about a Population Mean.

Okay, these are pretty cool, but how do they help students learn?  I often use a Demonstration in class to illustrate the topic of the lesson.  I hope that the animated graphic will be memorable for the students.  I also post a link to the Demonstration on the course web page.  I spend a few minutes in class describing how to download the plug-in, but most of my students are more familiar with downloading programs than I am.  Ideally, I also use a screen capture program, like Jing, to create a video of my computer screen while I use the Demonstration and explain the underlying concepts.  I end the video by instructing the students to explore the Demonstration on their own.  I make the video available on the course web page too.  My students use the video to review the ideas from the lesson.

I haven’t used assignments based on the Demonstrations, but it would be easy enough to write some.  (I would be more inclined to use this type of assignment in a computer-based course than in a traditional classroom setting.)  For example, to illustrate the roles of the various parameters in a hypothesis test, students could be instructed to set the Hypothesis Tests about a Population Mean Demonstration to examine a one-sided test with  and set all sliders to their far left position, except the sample size slider.  Set the sample size slider so the indicator is right below the  in the hypothesis test.  Now that everyone’s looking at the same illustration, students could be asked to explain various aspects of the illustrations.  For example, in your own words, describe what the purple are to the right of the red line represents.  What does the darker area to the right of the black line represent?  What does the area between the two lines represent?  Now move the slider labeled  to the right.  What happens to the graph?  Why?  How does this impact the -value?   Why doesn’t the black line move?  Return the slider labeled  to its leftmost position.  Now moved the slider labeled  to the right.  What happens to the graph?  Why?  What happens to the -value?  Why?  When does the conclusion change from “do not reject ” to “reject ”?  Why?  And so on.  Of course, these questions could be rewritten as multiple-choice questions if desired.

I enjoy offering my students a new way to visualize difficult topics.  The Wolfram Demonstrations Project makes it easy to offer students the opportunity to play with statistical ideas.