German Tank Problem
Resources:
        Activity Based Statistics p. 148-150
        Exploring Surveys and Information from Samples, Section IX, p. 75-83
        Statistical Reasoning, Gary Smith, p. 148-149

Included with this webpage:  Dot plot template, worksheet, template for numbered squares.

Subject: extending the idea of capture/recapture.

Introduction: During World War II, the Germans tried to make a complete tabulation of how much they were producing, but reports from individual factories were often late and not always reliable.  British and U.S. statisticians were keenly interested in estimating German war production, too, but they could hardly ask the German factories to send them reports.  Instead, they based their estimates on the manufacturing serial numbers of captured equipment (specifically the tire molds and tank gearboxes).  These serial numbers provided a sample that was very small, but reliable.  The records of the Speer Ministry, which was in charge of Germany's war production, were recovered after the war.  Special studies made after the war discovered that the British and U.S. estimates of German production were more accurate and timely than Germany's own estimates.

The challenge is to choose a good estimator for the total number of tanks.

Materials:  1 container per group (a wide mouth cup works well), numbered paper squares from 1-36 (see template below). Small dot-plot template.

Dot Plot template:  give one to each group.

  Group # _______     mean = _______           standard deviation = ______
 

    <----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o---->
            5     10    15    20    25    30    35    40    45    50    55    60    65    70    75

 

Clarification of ABS instructions:

1. Each group will get a cup with consecutively numbered chips in it.  DO NOT LOOK IN THE CUPS. Randomly select three chips without replacement.  Record the chip numbers, and return them to the cup.  Mix well and repeat several times.

2.  Each group should come to a consensus how they would use the data to estimate N.

3.  Each group will use their formula or rule, and 20 of the samples (size 3)  made by the class to come up with 20 estimates.  Each group will make a dot plot of their estimates, along with the mean and standard deviation, on the dot plot provided, and post it on the white board.  The class will study the different results and vote for the one which seems best.
Common suggestions:
 1. Double the mean of the three numbers.
 2. Double the median of the three numbers.
 3. Add the average gap between the three numbers to the largest number
 4. Add the smallest and largest numbers.
 5. Triple the mean or median of the three numbers.
 6. Use the largest of the three numbers.

4.  Somewhere along the way, plot one sample of three estimates and encourage students to look at the pattern.  Adding the average gap to the largest observed integer begins to make sense.
A fairly good estimator is:
N = [(n+1)/n]m     OR   N = m + (m/n)

Which is interpreted as adding the average size of the gap to the highest serial number.

N= true population (total number of tanks)
n = number of tanks captured
m = largest  serial number of the captured tanks