Category: Statistics

The unit on experimental design is one of my favorites in the AP Stats year, but the structure of a matched pairs experiment – where every subject participates in both treatments – often confuses students. For the past few years, I have been introducing students to matched pairs design through a sport which is sweeping America…

HALLWAY BOCCE!

In hallway bocce, students place two poker chips 5 meters apart in the hallway. Then, standing behind one of the chips, they roll a golf ball towards the opposite chip, trying to get as close as possible. With our carpeted hallways, the golf balls really take off, so some practice is needed to get the right touch. During this practice session, the students don’t know where this is all heading in terms of experimental design.

Next, the students are given a direction sheet for recording results. Each “stat-lete” is asked to play bocce 4 times, twice with their right hand, twice with their left, alternating hands. A coin is used to determine which hand to start with. Partners then measure their attempts and record results.  Note that today was “fashion disaster” day as part of our school’s spirit week.

Back in class, we then think about what could be conjectured before this experiment.  Sure, we could compare the attempts by right hands and by left hands, but what does this tell us?  We then settled on looking at players’ dominant versus their non-dominant hands, and made a dotplot of the results (note – my pre-made scale really was not sufficient here…those golf balls really fly!)

But this only allows us to compare hands in general. What we’d like to be able to do is determine if players are better with their dominant, rather than their non-dominant, hands. Subtracting these results, since all players participated in both treatments, allows for this comparison.

In the end, those reasults seem quite inconclusive, but that’s okay! Not all experiments prove conjectures, and we learn about the process.

Today’s class opener comes from my Advanced Placement Statistics class, but provides an important lesson for stats students of all ages.  A timeplot featuring two interesting data sets, and their changes over time is featured as students enter:

That’s quite a high r value we have for two variables, autism diagnoses and organic food sales, which would not seem so closely related. In conversation with the class we discussed the importance of clear communication, and how this article could easily be summarized and misinterpreted by our local newspaper:

ORGANIC FOOD CAUSES AUTISM, RESEARCH SHOWS

Uh oh….we have a problem.  And not an uncommon problem, as scientific studies which find correlations between variables are often misinterpreted as cause-effect studies.  The fun site Spurious Correlations by Tyler Vigen provides some wild examples of variables with strong (sometimes eerliy strong) correlations to help frame discussions.  Some fun examples –

• Divorce rate in Maine correlates with Per capita consumption of margarine (US)
• Worldwide non-commercial space launches correlates with Sociology doctorates awarded (US)
• Per capita consumption of chicken (US) correlates with Total US crude oil imports

Later, my students will be asked to read and respond to a “newspaper article” about a California school which analyzed their student data and found that student achievement correlates strongly to student height.  The school’s reaction to this correlation seems dubious at best, and with good reason….it’s a fictitious article I wrote symobolize the danger of seeking cause/effect from casual relationships.

A problem I gave as review for our statistics test today became not only a source of conversation regarding vocabulary, but provided me some insight into the problem solving approaches of my students.

Here’s the problem. A list of numbers is given, listed in order, with some numbers removed:

The list has the following characteristics:

• A mean of 76
• A range of 32
• An inter-quartile range of 21

Many students quickly understood the last blank must be 92, due to the range, but then became stuck.  As we’ve never explicity seen a problem like this before, the reactions from students was fascinating.  Some pockets of students had no fear in drawing circles and arrows to break down the data set. Others preferred to talk ideas out, but without putting pen to paper this doesn’t lead to solutions right away. I was thrilled to see a few students step up and take the lead, and explain their ideas to others, which then led to breakthroughs.  Identifying the positions of median and quartiles here lets us fill in one of the missing numbers:

But a subset of my class was content to watch from afar, waiting for hints which they assumed would come. Or worse, tuning out until I presented an explanation to the class….which never came.

And that last blank caused more trouble than I would have expected, as some students had trouble making the connection between the mean of a data set and the sum of its elements.  To help with this, I asked struggling students to provide me with any 4 numbers which had a mean of 10 (making them different numbers).  I asked students what I should be looking for to check accuracy besides computing the mean….and then, the light bulb!  All lists need to add up to 40!  So without explictly doing the empty blank problem in front of us, I sent students back to the board to think about this fact.  And the results were satisfying, as many of my fringe students could now complete the task and explain their procedure to their peers.

Students need to understand math ideas in many forms, and the concept of mean here demonstrates this need.  If you ask a student how to compute a mean, they most likely have little difficulty, and have had much practice:

Mean = sum of “scores” / count of “scores”

But in the missing numbers puzzle, the concept “felt” different and thus “new” to many students.  For me, this is where many students struggle in math classrooms.  Are we showing students how ideas and problems connect to big ideas?  Or does each combination of an existing problem become treated like a new experience?  It’s hard to break the pattern of students wanting specific rules for each type of math problem, when this is often the math conditioning they receive. But it’s worth the hard-fought battle.

And if you had fun with the challenge at the start of this post, try the similar problem I give later as an assessment: