Men with glasses poring over long columns of numbers. Tabulation of averages and creation of data tables. Lots of counting. The public image of statistics hardly corresponds to what OPIG do – even where OPIG’s work is at its most formally statistical.

OPIG exhibited a street stall at the Oxford Maths Festival to try to change that perception. How do you interest passers-by in real statistics without condescending and without oversimplifying? Data is becoming more important in the lives of all kinds of people and we need to be clear that it isn’t magic, but neither is it trivial. We need to prove that the kind of thoughtful reasoning that people put into managing their lives is the same kind of thing we do in data analysis.

Let’s look at one activity that OPIG did on the street at the Oxford Maths Festival.

The idea

We started with a compelling story: rumors were flying during the Second World War about how many tanks the Germans were producing. Allied intelligence needed to figure out if these numbers were true. In its simple retelling, the Germans simply numbered their tanks, but in truth there were two sequences of gearbox numbers, complicated chassis and engine numbers, and a set of numbered wheel moulds which left numbers imprinted in each of the wheels of the tanks.

However you parse it, the relevant problem is finding out how many gearboxes, chassis, wheels, or engines were in use, which tells you how many tanks are being produced, since no tank can have, for example, more than one engine. It’s a little hard to get close to a German tank, so Allied soldiers collected these numbers when they were destroyed.

This problem – finding the largest number of a range 1, 2, 3, 4, … N – is known generally as the German Tank Problem. Most mathematics educators have some familiarity with it, but on the street we found it works really well. This observation presumably speaks to the paucity of mathematics educators on the street.

How it works

The demonstration, while straightforward and quick, has a few subtleties. We cut open a box and cut 4750 slips of paper, numbered from 1 to 4750.

The first step is to ask how many slips of paper are in the box. Answers varied from one hundred to more than 10,000. Shaking the box helped encourage unreliable guesses and prevented people from reading the numbers off the slips in the box.

Then we asked people to pick one piece of paper out of the box and update their guess. We found a few aspects of this to be interesting. It is trivial to convince people that they will not get the top number in their guess. We all, then, have an intuition that the number drawn at this stage should be ‘somewhere in the middle’, which is completely wrong. There’s no particular reason to think that the guess should be in any interval over the distribution. It is true, however, that the mean of the first number chosen after many runs of the demonstration is 2375 and reasoning about average behaviour in this way turns out to be very powerful.

We then would ask people to draw up to four more slips. The point that people should absorb at this stage is that those guesses should assort evenly over the possible distribution, and you only need to add ‘a little bit more’ to compensate for this effect. We then precomputed the best guess for various numbers because the calculation is too tedious for a streetcorner – from which it becomes obvious that having more than five slips is nearly unnecessary to guess the maximum well. (See the histogram of estimates from five-slip draws below.) And 60% of guesses were slightly high of the true number, but the guesses that were too low tended to be much too low.

Going from blind indifference to a really solid guess is a powerful experience, and we can take people through it with every step of the reasoning on display. It shows what data analysis looks like at the research level and can be a great experience for the public.