54

Not a year goes by where we don't see at least one person Sorted into each House, and it's usually a fairly even distribution.

The Sorting Hat doesn't get a chance to study the mind of each child before Sorting, so it can't Sort them relative to each other (top X in smarts go to Ravenclaw, 4 bravest to Gryffindor, etc)

What would happen if a Sorting occurred where all the witches/wizards were extremely intelligent? Would they all get Sorted to Ravenclaw? What if they were all extremely brave? What if none of them were particularly smart or courageous, but they all wanted power?

We see that there's a fairly small number of beds per room in each dorm, would the Hat stop Sorting people into a House if it knew the House had no more beds? If so, could Harry Potter have ended up in Hufflepuff if his last name had been 'Zimmers'?

  • 1
    Magic™? I can't rule out someone picking up a clue from an interview or missed passage in a book so I'll leave this as a smart-alec comment rather than the half-assed answer it would be, but I bet that's what it ultimately boils down to, as far as in-universe explanations go. – Saiboogu Dec 16 '11 at 20:11
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    Hogwarts reconfigures itself in a couple ways (Room of Requirement, the moving staircases, possibly other things I'm forgetting...), so there's no particular reason the dorms would've been limited in size – Izkata Dec 17 '11 at 0:06
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    The dorm rooms don't seem to be very crowded. Harry sleeps in a four-poster bed. You could probably double the capacity by replacing them with bunk beds, and I think there'll still be enough space for the belongings. – b_jonas Jun 16 '12 at 11:29
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    I fear OP is too optimistic here. This falls so obviously in the category "author just build a story, not a setting" which I see no point in asking it at all. (+1 anyway because it would be a nice question, but doesn't apply to a non-setting such as this one) – o0'. May 28 '14 at 16:10
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    Hmmm I sense greatness in you, you are brave and clever, but quota for Gryffindor and Slytherin has been met. HUFFLEPUFF! – Zikato Aug 14 '15 at 13:28
29

It's hard to know whether there are house quotas without being able to compare the Sorting Hat's placements against the total number of students at Hogwarts. We don't know how many students are at Hogwarts or what their house placements are.

And, unfortunately, I think we have to take any proffered number of students at Hogwarts with a grain of salt, no matter the source. J.K. Rowling has ongoing discrepancies in her numbers when asked about it, and she admits she is "horrible at maths." At one point she put forth that there are approximately 1000 students at Hogwarts. If so, it would break down to approximately 250 students per house. There's a short article on how many students there possibly are at Hogwarts here. If we are to trust canon, the number of known students in each house does indeed seem to be evenly distributed.

However, I think, in your scenario, where all the students, or a disproportionate number of students, were suited to one house, then that is where they would go. I interpret what we know from canon to mean that the Sorting Hat sorts on ability, not on available bed space. So, no, there are not quotas based on house placement.

Canon doesn't demonstrate or state exactly how many beds are in each dorm room. The books are from Harry's POV; it may be that we only meet the Gryffindor boys and girls that Harry has the opportunity to interact with or notice; perhaps there are many more Gryffindors that Harry simply never notices or, more likely, mentions. In the article I linked to above, it's postulated that there are approximately 36 students per year, per house, which seems more reasonable, again based on Harry's point of view and what we see in canon (the books, not the movies; the movies show many more students per house than 36).

Neither canon nor J.K. Rowling adequately addresses how many students attend Hogwarts. The only answer we have is J.K. Rowling's estimate of "around 1000" students; however that number doesn't seem to be supported by canon or Harry's POV. In a nutshell, we don't know. Canon suggests an even distribution of house placements, however, regardless of the number of students the Sorting Hat sorts, which does not indicate a quota.

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    @DVK - While I won't argue that Crabbe and Goyle are the brightest bulbs in the box, I will quote the Slytherin welcome letter: Because you know what Salazar Slytherin looked for in his chosen students? The seeds of greatness. You’ve been chosen by this house because you’ve got the potential to be great, in the true sense of the word. You might see a couple of people hanging around the common room whom you might not think are destined for anything special. Well, keep that to yourself. If the Sorting Hat put them in here, there’s something great about them, and don’t you forget it. :D – Slytherincess Dec 18 '11 at 19:00
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    Well, I suppose winning a Darwin award is rare enough that one might qualify it as "greatness"... :=D – DVK-on-Ahch-To Dec 18 '11 at 20:25
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    When it comes to Crabbe and Goyle I can't help but agree (even if I'm not supposed to :D ) Well, I suppose they have to have something to cling to . . . the knuckle-dragging cretins that they are ;) – Slytherincess Dec 18 '11 at 20:46
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    @DVK: Crabbe, despite being seen as a dullard and little more than a goon for the better part of 6 books, mastered multiple powerful Dark curses and learned to use FiendFyre before becoming a legal adult. – Jeff Aug 6 '15 at 13:55
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    If anyone ever wondered how dangerous Gregory Goyle was; after leaving Hogwarts, he became a MMA fighter! bbc.co.uk/newsbeat/article/36137321/… – maguirenumber6 May 1 '16 at 16:54
25

Not a year goes by where we don't see at least one person Sorted into each House, and it's usually a fairly even distribution.

Let's do the math. Assume there are N students per year, and the probability for any one student is always 1/4 to be sorted into house h, where h ∊ {G, S, R, H } =: ℋ. Define "a fairly even distribution" as

nminnhynmaxh ∊ ℋ, y ∊ {1991, ... 1997}

where nhy is the number of students in house h in year y. The probability for this is Py7, where Py is the probability that any one year is evenly-distributed. That value is calculated by the Haskell program below (it's not very well done, essentially brute force).

To get a result, we need to know the number of students in each year. Let's first try the estimate by the article Slytherincess already linked to, i.e. 10 students per house per year / 40 in total. Allowing for a range from 6 to 16 students per house, we find out

GHCi> ( p_YearHasFairlyEvenDistrib (6,16) 40 )^7
0.22543290063072918

that the probability is only 22.5% that Harry will never have observed a year with not-fairly-even distribution. But it becomes quite a lot bigger if we allow just a slightly larger margin

GHCi> (p_YearHasFairlyEvenDistrib (4,20) 40)^7
0.8725318786933933

makes 87%! Now, only 4 students per house can't really be called even distribution anymore, but I don't think we can prove this never happened in the course of the books.

If we rather use the number of students JKR herself gave, 1000 in the whole school ⇒ 143 per year (the program as given below won't do that, at least not within 8 GB of memory – I had to optimize it a little) ⇒ on average 35 students per house, we can restrict ourselves to the substantially more even-looking range (25,50) and still get a probability of 66%.

So, all in all, it's really a question we can answer with Hogwarts doesn't really need quotas; even with perfectly equal treatment of all students there will very seldom be a problematically uneven distribution of students in the houses.


import Data.List

data HousesDistrib = HousesDistrib { studentDistribution :: (Int,Int,Int,Int)
                                   , distribProbability :: Double
                                   }

instance Show HousesDistrib where
  show (HousesDistrib d p) = " " ++ show d ++ " @" ++ show p

studentIntoHousePossibilities :: HousesDistrib -> [HousesDistrib]
studentIntoHousePossibilities (HousesDistrib (g,s,r,h) p)
   = [ HousesDistrib (g+1,s,  r,  h  ) p'
     , HousesDistrib (g,  s+1,r,  h  ) p'
     , HousesDistrib (g,  s,  r+1,h  ) p'
     , HousesDistrib (g,  s,  r,  h+1) p'
     ]
  where p' = p/4

summarizeEqualDistribs :: [HousesDistrib] -> [HousesDistrib]
summarizeEqualDistribs = map sumup . groupBy distribEquals . sortBy distribOrdering
  where sumup = foldl1' (\a b -> HousesDistrib
                                  (studentDistribution a)
                                  (distribProbability a + distribProbability b) )
        a`distribEquals`b = (studentDistribution a == studentDistribution b)
        a`distribOrdering`b = compare (studentDistribution a) (studentDistribution b)

allPossibleDistribs :: Int -> [HousesDistrib]
allPossibleDistribs n = distribSequence [HousesDistrib (0,0,0,0) 1] !! n
  where distribSequence = iterate ( summarizeEqualDistribs
                                   . (>>=studentIntoHousePossibilities) )

allFairlyEvenDistribs rng = filter (isFairlyEvenDistrib rng) . allPossibleDistribs

isFairlyEvenDistrib (nmin, nmax) (HousesDistrib (g,s,r,h) _)
    = ok g && ok s && ok r && ok h
  where ok n = n>=nmin && n<=nmax

p_YearHasFairlyEvenDistrib rng nStudents
   = sum . map distribProbability $ allFairlyEvenDistribs rng nStudents
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    Perfectly correct, mathematically...and utterly pointless since the Sorting Hat doesn't decide at random. Plus, Harry only sees something like 3 Sortings (including his own) in the 7 books. – Jeff Dec 18 '11 at 1:33
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    @Jeff: of course the Sorting Hat doesn't decide at random, it's the students I assume to be randomized. – leftaroundabout Dec 18 '11 at 1:38
  • @Gallifreyan: hm, that must actually have been one of the first Haskell programs I ever wrote... in the years since, I've been doing it whenever possible. – leftaroundabout Feb 20 '17 at 19:10
  • I wish my first Haskell program would compile :D Except Hello, World!, of course – Gallifreyan Feb 20 '17 at 19:13
  • Have a [rather belated] upvote! Feel free to put that code up on Code Review - with a clever title I predict this could easily turn into a Hot Network Question... and get optimized to death! =) – Mathieu Guindon Jul 10 '17 at 18:49
5

I wouldn't assume that there's always a small number of beds in each room - that's probably something that changes to match the number of students who get sorted. Keep in mind that the war with Voldemort seems to have had a massive number of casualties, cutting down a huge percentage of the wizarding population in England. It seems likely that before this generation the number of students was much larger, meaning that there would've had to be more beds, and also making it far more statistically improbable that you wouldn't have at least a few students in each house.

Also, it seems extremely common for young wizards and witches to be sorted into the same houses as their parents, presumably because they are raised to value the same things that their parents value. I'm sure this also contributes to the unlikelihood of having a 100% Ravenclaw year.

4

You're right that the Sorting Hat can't sort them relatively, and I think that's a hint that there are ability/potential cutoffs, not quotas.

The beds won't be a problem, I'm sure the rooms could magically expand and beds could be moved around or created as necessary; the school is magical and the teachers and headmaster are all quite powerful magicians themselves, after all.

I think a fairly even distribution would be fairly self-perpetuating, since houses tend to run in families (though we know that's not always the case). And the founders probably picked similar numbers of students for their houses while they were choosing for themselves. And with ~17 students of each gender per house per year (1000 total, 7 years, 4 houses, 2 genders; Harry's class was particularly small), it would be subject to some statistical fluctuation, but unlikely to result in a greatly lopsided distribution.

So I don't think they have quotas.

And I think the strongest evidence of this comes from JKR's assertion (quoted in slytherincess's answer) that the sorting hat has never been wrong. If there were quotas, it's unlikely it could have gone over a thousand years without being forced to put someone in a house because of a quota instead of it being where he or she belonged.

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