“It is difficult to count objects that are moving around”
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Whether it’s enemy tanks in battle, animals in the wild or cutlery in a busy canteen, it is difficult to count objects that are moving around. Luckily, there is a technique that can estimate how many there are of something without requiring you to count every single one.
The capture-recapture method involves getting a sample – waiting for some animals to wander by, for instance, then collecting some – marking the individuals distinctively, then releasing them back into the population. After some time has passed, you repeat the process to pick another group of animals and count how many of them are already marked.
If you captured, say, 50 animals initially and marked them all, then on your recapture step you found half the animals you saw were marked, this tells you something about the whole population. Since half the sample is marked, this implies that half of the whole population is marked – so there must be about 100 individuals. This can give a reasonably accurate estimate of a population, without having to find and count every single member of it.
During the second world war, allied statisticians wanted to determine how many tanks the German army was producing. Captured tanks couldn’t be re-released, but, as tank components are marked with serial numbers, another approach allowed them to make an estimate. They logged the serial numbers of all captured or destroyed tanks, working on the assumption they were numbered sequentially and randomly distributed. If the largest serial number in your data is L and the number of captured tanks is n, one estimate for the total number of tanks is given by L + L/n.
So, if we had four numbers, the largest of which was 80, we could assume the whole range extends about another 80/4 = 20, so there would be about 100 tanks overall. This is known as the German tank problem in statistics.
One of my favourite population estimation stories was told to me by a teacher friend, who tasked her students with estimating the number of forks in the school canteen – impossible to count as, at any given time, a number will be in use and others will be in the wash.
Her class “captured” a set of forks and marked each one with a drop of nail polish, then released them back into the population. A week later, they recaptured another sample population and used it to make an estimate of the total number of forks.
Researchers performed a similar experiment 20 years ago. A worrying number of teaspoons were going missing in their lab, so they marked a set of spoons before releasing them, studying their movements and publishing the results. It turns out science is effective: the publication of the paper did result in five teaspoons being sheepishly returned by spoon stealers in the building.
Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks
For other projects visit newscientist.com/maker
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