The German Tank Problem with Fathom
This document contains the Fathom instructions for the German Tank problem as I do it. It is not a
plan for the entire activity, which begins with a good deal of discussion and drawing counters from a
bag.
In a new Fathom document, we need to create a population of tanks. Drag a New Table from the
Object Shelf to the workspace. Name the attribute “TankNumber.” Because each row in a case table
is numbered with a caseindex, the formula
TankNumber = caseindex
will create of a list of integers, 1, 2, . . ., n. Single-click on the word TankNumber to select the
attribute and from the Edit menu, select Edit Formula. Enter the formula “caseindex.”
This creates a formula to determine the values for any cases (tanks) you have, but has not actually
created any tanks. To create tanks, select New Cases under the Collection menu. A gold box (the
collection) filled with little spheres will appear on the screen. In the dialog box that opens, type 342.
This will create an army of 342 tanks.
The next couple steps are not necessary for the simulation to work, but are helpful from a
pedagogical standpoint because they help students understand the simulation. First, under the gold
box, double-click on the name “Collection 1” and, in the dialog box that opens, rename the
collection “Tanks.”
Next, select the collection and drag the lower right corner of the blue frame so you can see the icons.
Each ball represents one tank, and you should note that each tank is labeled “a case.” It is more
instructive to have each labeled with its number.
Click on the column title in the table and, from the menu, select Table | Use as caption. This will
number each icon with its tank number.
Next, we’ll select our seven “captured” tanks. To select a sample, single-click on the collection so
the blue frame appears. Then choose Collection | Sample cases. A new blue box will appear, and
you should see a blue ball pass from the collection box to the sample box to show that a sample is
being selected. Single-click on the sample box and then drag the lower right corner of the frame until
you can see all 10 balls in the sample. (Ten is the default sample size.) Now you can see why we
changed the captions: It is much more instructive to be able to see which tanks were selected when
we do the simulation.
Notice the screen layout. The screen will quickly become crowded. One way to keep things
organized is to keep the tank army on the left, the sample in the middle, and leave room at the right
for keeping track of our sample statistics!
To change the size of the sample, double-click somewhere inside the sample frame to open the
inspector for the sample. Change the number of cases to 7, turn off With Replacement, and click on
Sample More Cases as in Figure 11 on page 51.
Now you can see the sample of seven tanks. Click the Sample More Cases button above the sample,
and a new set of seven tanks will appear. I typically encourage students to do this several times to
see the repeated sampling.
In the classroom, students can get lost in the steps of this process so it is good to remind them the
purpose of our activity, and links between this and the physical simulation they just did.
Some typical estimators were introduced earlier. Students can calculate one or all of these
estimators, or others they create, for each sample. To do multiple estimators, open the inspector for
the sample (double-click somewhere in the sample), and click the Measures tab. Then enter the name
of the estimator, such as “Double_Mean” for double the mean. (Fathom does not allow spaces in
names of attributes or measures.) Double-click in the cell for the formula and type in
2*mean(TankNumber).
Note that the name of the estimator can be different, but there are restrictions on the symbols that
can be used. The formula is strict in its syntax. Some other common estimators, with formulas, are
shown below. The estimates for the current sample of seven tanks are also given in the Value
column.
To generate results from many samples, be sure that the Sample of Tanks collection is selected.
Select Collection | Collect Measures. A third collection box will open. This contains a case for each
sample of 7 that comes up. In the inspector for Measures from Sample of Tanks, change the number
of measures to 500 or so. Deselect “Animation on” and select “Replace Existing Cases.” Click
Collect More Measures.
Drag a new graph from the object shelf. Drag the name “Double_Mean” to the horizontal axis. Then
drag each of the other names to the axis, dropping it on the “plus” sign that appears. This will stack
all the different plots.
To see how well each estimator works, select the graph and select Graph | Plot value. Type 342 in
the formula editor.
It’s pretty striking now that some methods work well on average and others do not. To make this
more clear, plot the mean of each distribution. (Plot value with the formula mean( ).) This brings up
the idea that we want an unbiased estimator. You can re-order the plots by dragging the names on
the left. Drag the biased estimators to the bottom.
Now, out of the unbiased estimators, it’s easy to see which have the smaller spreads.