On the mean, your estimate based on the number of Portkeys should be correct. There's another limiting factor that comes into play: the carrying capacity of a Portkey.

Even if there are certain areas (e.g. Hogsmeade) with much higher densities of wizards, the individual Portkeys can only take a limited number of people. You can only crowd so many people around the head of a statue, or a boot, or an old football.

So, areas of higher density will simply have a lot of Portkeys within walking distance. Areas like Ottery St. Catchpole will have a single Portkey that serves three or four families' worth of ticketholders. And then there will probably be some areas of lower density, where you can't fill a Portkey's capacity without certain wizards walking more than a day. I don't know what you'd do here - honestly it would make more sense to have them converge on Diagon Alley and have the Portkey depart from there.

That said, there's also the question of people who don't need Portkeys to get to the World Cup. Many ticketholders simply Apparated to the Cup; we don't know how many did this, but it must be a considerable number. So you could make a fudge, and use this to increase your estimate of the population density of wizards in the UK.

 

Just to make a point regarding the 'Portkey Calculation', Amos and Arthur make mention of the Fawcetts and Lovegoods. These are both families with children young enough to be in Hogwarts, thus unable to apparate and as such have the requirement of a portkey to travel to the World Cup.

This is evidenced by a) Luna and b) that a Fawcett was admitted to the hospital wing at a later date with a long white beard having attempted to bypass Dumbledore's age-line.

I can't be bothered to work out if this significantly influences your calculation or point, but I felt it was worth observing.

EDIT: Of course, working from this basis, I'm going to make some other hugely assumptive pieces of calculation, but first, it requires some rationale.

I'm going to take the seven children (Lovegood, Fawcett, Diggory and the four Weasleys) in that 1250km² that were eligible to attend Hogwarts at that precise moment in time as representative of the wizarding world at large. This may perhaps seem like a stretch as the Weasleys are often remarked of as having a remarkable number of children for a wizarding family, however I'm also basing this on the fact that most 'normal' families that produce wizarding children are presented as having more than one, but fewer than seven children. Those characters who we know are an only child have almost invariably had something terrible happen to their parents (Harry, Neville, Luna) or have families that are in other ways abnormal (Dean Thomas, Lupin, Malfoy, etc) or have had magical siblings written out of the books (Hermione).

So assuming four families were to occupy our hypothetical 1250km² area, with an average of two hypothetical children each, we come to eight.

With this in mind, multiplying these figures by 200 (to make up the rest of the size of the United Kingdom), we come to a number of children at Hogwarts of between 1400 and 1600 children.

Yeah, that's how absurd trying to rationalize JKR's numbers can be c:

 

Portkeys are simply mass transportation, whereas apparation is at most three people (the apparator and two clinging to the arms). Where you're expecting to take more than a couple of people, portkeys are provided. Just think of them as a bus to apparation's taxi.

 

If it is this simple, why didn't Amos apparate Cedric to the World Cup? (I mean ignoring the obvious; that JKR clearly hadn't thought of the idea by Book 4)

 

Because some people prefer to travel with friends? Pass, could be anything.

 

It could be anything from:

Fancying a hike, trying to get into the 'matchday spirit' (I know that the last time I went to Wembley, my dad made me do all sorts of silly rituals), Amos couldn't side-along Apparate, Cedric (or Amos) wanted to meet Harry, they wanted a chat with the Weasleys, they planned to get blind drunk and you had to take a portkey there to take one back (plausible, considering the Ministry's beurocracy), or perhaps just stupidity, an apparently common afliction in wizard-land.

 

Or because JKR wanted to introduce Cedric as a major character and that was the easiest way to do it.

 

Isn't that a bit like the whole 'Harry meets the Defense teacher each year before school' thing? Introduce Amos so they have a more stable wizard to talk to when the Dark Mark scene happens. Reintroduce Cedric since now he's gonna have a bigger part in the story than just Hufflepuff's Seeker. Enter in Viktor Krum as well so we know him a tad bit. And I think it might be safe to assume Fleur was in that huddle of Beauxbatons girls in the forest and they just weren't paying attention 'cause of the Death Eaters. Now not only has Harry heard Crouch Jr., heard of Moody, but he's also 'met' all the future Champions. That would certainly keep with Rowling's way of tossing someone at you that becomes part of that years story. i.g. Professor Quirrell, Lucius Malfoy, Padfoot

 

The "bloat" of the Ministry can be explained by the fact that the magical government is engaged in the giant, never-ending project of enforcing secrecy. I should imagine that maintaining secrecy takes up a large portion of wizarding industry.

Think back over what we know of canon. Beyond the obvious (Obliviation squad) you also have significant portions of various departments engaged in maintaining secrecy.

Department for Control of Magical Creatures - Fantastic Beasts and Where to Find them tells us that part of their function is to keep magical creatures a secret.

Department of Magical Games and Sports - GoF tells us about all the secrecy charms they put on the Quidditch Stadium, QttA talks about various regulations that the Ministry enforces to keep Quidditch secret.

Department of Magical Law Enforcement - Arthur spends a fair amount of time making sure magical items don't get into Muggle hands.

Department for Magical Accidents and Catastrophes - Significant;y devoted to maintaining secrecy, e.g. covering up accidental magic, responding to events like the dragon sighting at Ilfracombe (to which they responded very slowly).

 

Well partially that, but also consider how difficult it would be for a benign Ministry to control a population of wizards who felt no reason to follow them.

After seeing Ogden's memories in HBP I've always considered the Ministry and it's relative size in comparison with the wizarding world a reflection that without the majority of wizards fundamentally tied up into the infrastructure of their own government and thus given some accountability for their actions, the Ministry couldn't work.

 

This is the only estimate I'm actually aware of that JKR has made in terms of estimating the magical population. If we take those figures as canon, that means there is just one wizard or witch for every 20,000 muggles in the UK; in which case it's not really a surprise they've been able to uphold the Statute of Secrecy so easily.

 

I like how JKR continually admits that she didn't put much thought into this stuff, and people still try to extrapolate the world from the smallest bits of information.

 

At least she doesn't try and pretend that she's got it all figured out.

 

An unfortunate part of a fanfic writer's job.

 

That must be really frustrating for other fantasy/sci-fi writers. You know, the ones who agonize over numbers and making sure their worlds work? To look at JKR's billions and then at their own bank accounts must be pretty demoralizing and infuriating, regardless of what they saw publicly.

 

Honestly, I think getting too caught up in the details sometimes is a bad thing. I mean, this is fiction after all - as long as the reader can suspend their disbelief, the plot and writing are far more important. JKR's details might be weak, but she wrote a pretty damn good story.

But yeah, you should still probably know the rough population of the society you're writing about, or at least the amount of students at the school 90% of your book is set in though.

 

That's the thing though isn't it? Since the books are from Harry's point of view and he doesn't take a large interest in learning everything he can about the Wizarding World or at least Britain. We only know the students and adults he hears about or meets.

But Jo can flash a rough draft list of Harry's year and we all latch onto it with an iron grip 'cause we want to know more about the world Harry is in. Which makes me wonder if Rowling has a rough list of Katie's year, Fred & George's year, Luna's, etc., etc. :awesome

 

I wasn't really speaking about Harry not knowing the populations - it's well established that he isn't a reliable narrator, etc. I was talking about Rowling's interviews about the numbers being vague and contradictory. She clearly doesn't know the answer, and never considered it too closely when writing the series.