A population census starts from an address register: a census taker visits every address in this complete register to check how many people live there. The number of pets can be counted in the same way. But how can we count wild animals?
The basic difficulty is that, by definition, we cannot "count" a population of wild animals when we cannot observe every individual at the same time. How many squirrels are there in Central Park? How many wild boars in the Ardennes forest? How many Arctic char in Lake Annecy? We cannot know exactly, but we can estimate the numbers. One way to do this is with a statistical method called capture–mark–recapture.
Capture–recapture methods ---------------------------------
The method begins with the capture phase: c individuals are captured from the population under study (without harming them!). Next comes the marking phase: the captured individuals are given a durable marker that does not interfere with them. They are then released back into their original habitat, where they mix with the population as a whole. We then conduct a recapture: we sample r individuals from the population and see how many were marked in the first capture. There are m of them. The total population N can then be estimated as N^\widehat{\mathrm{N}} by equating the proportion of marked individuals in the sample (m / r) with that in the population (c / N^\widehat{\mathrm{N}}).
We have mr=cN^\frac{m}{r}=\frac{c}{\widehat{\mathrm{N}}}, or equivalently N^=crm\widehat{\mathrm{N}}= c\frac{r}{m}.