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If we assume that Harry's year in Hogwarts is an average size, and that each house gets roughly a quarter of the amount of new students each year, then it seems to me that Hogwarts doesn't house anywhere near enough students to cater for the wizarding population of Britain.

In Harry's school year, the male portion of Gryffindor consists of Harry, Ron, Neville, Seamus and Dean. If there are roughly that many girls in that years Gryffindor (I can't remember anyone other than Hermionie right now), then that makes roughly 10 in each house each year, which makes roughly 40 students a year. Over the 7 years, that makes about 280 students in Hogwarts at any one time, roughly 70 students per house.

This doesn't anywhere near high enough to teach all of the wizarding populations children, which although considerably smaller than the Muggle population, is still considerable, as evidenced by the size of the Ministry of Magic and how full Diagon Alley always seems to be.

Has this ever been explained? Out of universe, it seems that J K Rowling is just bad at maths, but in universe?

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    "Oh, my dear maths!" – SQB Jun 11 '14 at 12:02
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    "I hoped no one would ever notice!" @SQB – CandiedMango Jun 11 '14 at 12:05
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    @DrRDizzle: Maybe they have just not been mentioned just like possibly all the other ones. I don't believe the whole planet only has three of them. I am very confident that there is a school full of stubborn scottish wizards ^^ – PlasmaHH Jun 11 '14 at 13:03
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    We're also assuming that there's an equal distribution between houses, which may not be accurate. Hufflepuff could well be a much larger House than the others because it doesn't really emphasise a particular character trait; a slightly harsh interpretation is that it's the House for the rejects who don't fit anywhere else. All of that said... JKR is notoriously bad at math, I'm not sure we'd ever be able to truly resolve the discrepancy to a satisfactory manner; the numbers just don't match up, there's nothing to be done about it. – Anthony Grist Jun 11 '14 at 13:42
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    @Kevin It's a shame that their seems to be no in-universe answer. The wizarding population DOES seem to fluctuate massively over the course of the books, whereas logically it should only be growing (year on year, not over the course of the books) - Muggleborns seems uncommon but not rare, whereas I can only recall one Squib (Filch) in the entire series, and we know that Wizards marrying Muggles is increasingly common, and they seem to mostly produce Wizarding children. It seems to me that magic is a dominant gene, not a recessive one. – Dr R Dizzle Jun 11 '14 at 13:51
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For this we need to look at the following things -

  1. Is Harry's class size indicative of average number of annual wizardly births?
  2. How large is the population of Wizarding UK anyway?
  3. Why does Diagon Alley always seem to be full?
  4. What's with the number of people in the Ministry?

Is Harry's class size indicative of average number of annual wizardly births?
This seems unlikely. Since Harry was born smack in the middle of one of the worst conflicts Wizarding UK faced. We know that people were disappearing every week for around 11 years, for a low estimate of around a 1000 adults people dead - which would be at least 10% (but more likely a lot more) of the population. Not only people too dead to spawn kidlets, but also not exactly an ideal time to be deciding to have babies (or be pregnant) anyway.

How large is the population of Wizarding UK anyway?
We have reason to believe average wizarding lifespan to be around 150 years; and assuming a conservative average annual birthrate over those 150 years to be around 50 gives us a population size of 7500. Which is not a huge number, but not exactly a tiny one either. Also consistent with comments by JKR.

Why does Diagon Alley always seem to be full?
Diagon Alley seems to be a street market style affair - and it definitely doesn't seem to be very large from what is described in the books. It wouldn't really take a very large number of people for such a place to appear crowded. But in addition to that, there's also the fact that Harry (and hence the readers) tend to see DA during what would possibly be one of the busier portions of the year for the market - the back-to-school shopping season. All those kids buying school supplies has got to create a bit of a crush.

What's with the number of people in the Ministry?
There seems a very large number of people in the Ministry compared to the total population, that's fairly true. But our window into the Ministry is through the Weasleys and the OoTP. Given that Arthur is a prominent civil servant and the OotP by its very nature would be made up of people likely to be Aurors or involved in politics, that seems at least one easy way to answer why so many of the adults we see, have something or other to do with the Ministry.

But in addition to that, there's also the question of the makeup of wizarding society. It's not a completely isolated population - it's a secret population within a much larger one. Most of the functions of the MoM seem to do with maintaining secrecy - so that's an extra function of the government that soaks up numbers - most of the Department of Magical Law Enforcement and Department of Magical Accidents and Catastrophes. Then there's also the Department for the Regulation and Control of Magical Creatures - given that these people need to deal and liase with whole societies of sapient magical people in addition to regulate dragons and other non-sapient magical creatures of that sort - their numbers being more significant than their Muggle counterparts doesn't seem too far fetched either.

Edited to add some numbers on Hogwarts and Wizarding Britain numbers from Jo: I found this interview from JKR regarding Hogwarts numbers:

Then I’ve been asked a few times how many people and because numbers are not my strong point, one part of my brain knew 40 [original characters she had created for Harry's year], and another part of my brain said, “Oh, about 600 sounds right.”

600 in Hogwarts doesn't seem so small after all. But that paints Harry's year as being more than half the usual yearly intake ((600-40)/6) of around 90, which then goes on to give up a total population size of 13,500, take around a couple of thousand for the casualties of the two Wizarding Wars in the last century, and that's still a fairly decent number.

But then that same interview goes on to say (in the very next line):

Let’s say three thousand [wizards/witches in Britain], actually...

What?! That means a Hogwarts year should have no more than 20 students at most! Harry's year is overpopulated; and total Hogwarts population should be something like a 150; and casualties under Grindelwald and Voldemort were massive percentage chunk of population. Or maybe most Wizards only live up to a 70-ish, same as Muggles, except a lucky few. That's the only way to explain those numbers.

Thankfully, JKR gives us an out and ask that we: "don’t hold me to these figures, because that’s not how I think."

In Conclusion: A 'corrected' average Hogwarts size of around 400-450 might be a good compromise between JKR and McNumbers. Harry's year in pretty small with 40 students, but the average yearly intake is 60-65 (two dorms per year per house per gender with 3-4 kids each (explains MWPP), instead of one with 5); leading to a total British Wizarding population of around 10000. The rest of the points in my answer still apply, I think.

And the explanation for this, as given by JKR herself:

Oh, Emerson, my maths is so bad.

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    This is actually a great answer. I still feel like the wizarding world in the books is portrayed as too large for the amount of wizards that there are according to the evidence/explanation you provided, but this clears up a lot. – Dr R Dizzle Jun 11 '14 at 18:00
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    @DrRDizzle considering Diagon Alley is a popular trading point, and wizards have instantaneous travel for miles and miles around. They don't have to drive 3 hours to get to it. It skews things from our Point of View. – user16696 Jun 12 '14 at 4:55
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    @cde Yeah, I hadn't considered the fact that Diagon Alley and Hogsmeade might be the only wizarding areas in the entire of the UK. Between portkeys and apparation, why set up shop anywhere else? – Dr R Dizzle Jun 13 '14 at 8:03
  • On the number of Ministry of Magic employees, note that there were 500 people put solely on building the stadium for the World Cup (not including other staff which were involved in those preparations), which suggests a total workforce of some thousands. I'd guess the MoM had a workforce of closer to 10,000. – The Dark Lord Jul 28 '17 at 19:00
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There was a presumed baby boom after and during the early stages of the First War with Voldemort, thus the number of student would be disproportionate to the number of adult wizards. Including muggleborns, the student body is not a guaranteed comparison to the adult population.

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    Got a source on that? – Gallifreyan Dec 24 '16 at 15:27

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