In preparation for the biggest first-edition press run in the history of the world, I'm rereading all of the first six Harry Potter books. While doing that, I came up with a question that's not answered adequately in the books: Just how many wizards are there?
While we can't get an exact answer, we do have enough information to allow a Fermi estimate of the answer. (Named after Enrico Fermi, a Fermi estimate attempts to get to within an order of magnitude of the correct answer to a problem based on a limited amount of information and reasonable estimates of the rest.)
The crucial fact we know is that Harry Potter's boys' dormitory room in Gryffindor has five beds. Assuming that the girls' dorm has the same number of beds, and each of the four houses has the same number of students, we see that Hogwarts admits forty students a year. We know of three schools of wizardry in the world, so that means that there are 120 wizards of a given age, all throughout the world. Hogwarts itself would have something less than 40×7 = 280 students, since many students leave before the end of the seventh year.
Of course, not every wizard is admitted to a school. We know of some wizard family members, such as Fitch, who have extremely limited powers. Presumably they aren't generally admitted into schools — though the admission of Neville Longbottom must mean that the schools are willing to scrape near the bottom of the barrel, at least for legacies. Then, too, an unknown number of Muggles are admitted to the school.
Let's assume, for the sake of argument, that the schools admit ten percent of all who could conceivably be called wizards. This is probably too low a figure, so our final number will be on the high side. Since the schools admit 120 students a year, there are approximately 1200 wizards of a given age, in the teen years, at least.
We then assume that a typical wizard lives for about 100 years — if they don't live longer than us, then they aren't really trying, after all. And we know that Dumbledore has been a teacher at Hogwarts for at least fifty years, so he's at least seventy, and showed no sign of slowing down until his untimely misfortune in the sixth book. One hundred years is about right, I'd say.
If we then take 100×1200, we get a really rough estimate of 120,000 wizards in the world. This is probably on the high side, given the Neville problem and the fact that many wizards will die before they get to 100.
One hundred thousand wizards in the world, give or take. This is a bit small, I'd think, for all the wizards that populate the books, including complete sets of professional Quidditch leagues. So let's see if we can increase the count.
Apparently, J. K. Rowling once said that Hogwarts has about 600 students. This means that Harry's class is a aberration, in that it's about half the sizes of an average class, but then Harry was born during the Voldemort years, when the birthrate was no doubt rather low. In this case, each school admits on average about 100 students per year, so our numbers go up by about a factor of 3, to, say, 350,000 wizards in the world.
That's still too low, probably because the books are Euro-centric: as far as I recall, every character is from Europe or the Near East. In particular, there aren't native Asian or North/South Americans in the books. So let's assume that the three schools mentioned draw from Europe (including European Russia) and Northern Africa — a population of roughly 500 million, one twelfth of the world's total. Presumably there are then are about 33 other schools in the world to teach the other wizards, or maybe the rest of the world is run by the White Council.
So we must multiply the 350,000 answer above by 12, which gives us a total of about 4 million wizards in the world, and that's probably an upper bound.
And that's not a bad answer. It's big enough to form a society (think of the Jews in pre-war Europe) that interacts with the rest of the world but keeps itself separate. And, as you'll note if you've been following my arithmetic, this is an order of magnitude calculation, meaning that there could be as few as 500,000 wizards in the world, and as many as 40 million. I'm pretty sure there aren't more than 40 million wizards in the world, or we'd see more of them.
Four million wizards, give or take, and they'll all be ahead of you in line on July 21.