Bring Back the German Tanks

Today's blog post comes from AP Stats guru Josh Tabor. Josh is an AP Statistics teacher at Canyon del Oro High School in Oro Valley, AZ. He is a long-time AP Reader, Table Leader, and Question Leader and presents workshops around the country. He is also the co-author of The Practice of Statistics for AP Statistics and two textbooks for on-level statistics: Statistics and Probability with Applications and Statistical Reasoning in Sports.

With your sampling distribution unit on the horizon, it’s time to revisit and revise the old German Tank activity.

I love this activity for a number of reasons, including the fact that it helped me understand the idea of a biased estimator after years of struggle. I also liked the creativity involved and how it gave students exposure to sampling distributions other than sample means and sample proportions. But, judging from reactions I get at workshops, many teachers have tried—and subsequently ditched—a version of this activity. If you aren’t familiar with it, here is a quick summary of the context and the activity.

The Original Activity

In World War II, the Germans were meticulous about putting serial numbers on every piece of equipment they produced. When the Allies captured a German tank, they wrote down the date, location, and every serial number they could find. This information was passed along to statisticians in Washington D.C., where they used the data to estimate how many German tanks they might encounter on D-Day. [i]

A common version of the German Tank Activity starts with a bag of objects numbered from 1 to N, where Nis the number of objects in the bag. I use craft sticks and call it the “Craft Stick Activity.” I don’t tell students about the German Tank context until the end, as solutions to the activity are easily searchable otherwise.

With students, I first present the bag and pull out a stick or two to demonstrate what’s inside. I then explain that their goal is to estimate the total number of sticks in the bag, based only on a random sample of n = 7 sticks and the fact that they are numbered from 1 to N. I don’t give them the sample of sticks until the end of the activity, as students tend to obsess over those 7 stick numbers rather than think about estimation methods more generally.

In the traditional approach to the activity, students work in groups to brainstorm statistics they could use to estimate the total number of sticks in the bag. I usually offer an example to get them started, such as doubling the sample median.

And this is where the activity typically grinds to a halt. Students—even working in groups—have a very difficult time creating their own statistics. The frustration they felt overwhelmed any good that came out of the activity.

What I Do Now

Then I met David Ellinger. David and his students at Seattle Preparatory School had the same frustrations as we did. But instead of just struggling with it year after year, he improved the activity by asking “what do I really want kids to gain?” He decided the most important goals are to increase student comfort with sampling distributions and for students to discern key features of sampling distributions, including the bias and variability. He simplified the activity to focus on student evaluation of various statistics rather than trying to create the statistics themselves.

Inspired by David’s realization, here is how my updated activity works:

  • Students form groups of 3 or 4 students.

  • Each group is assigned a preliminary statistic to use. I only use three different statistics at this point, so some groups end up with the same statistic. The three preliminary statistics I use are:

a. Twice the sample median

b. Sample mean + 3 sample standard deviations

c. Sample maximum

  • To see how well each statistic works, each group tests their statistic by assuming the population has N = 100 sticks. Using their calculators, they generate a random sample of n = 7 numbers from 1 to 100, calculate their statistic, and record the value. They do this as many times as possible, adding their values to class dotplots. Here is an example of what the dotplots might look like.