This question already has an answer here:

Simply put, every House has some characteristics they search after. Gryffindor has bravery, Slytherin has purity, and Ravenclaw has intelligence. But Hufflepuff?

Said Hufflepuff, ‘I’ll teach the lot, And treat them just the same.'

Hufflepuff takes students from far and wide (ideally), as there are no special requirements for being chosen. Thus, logic would dictate that there would be more Hufflepuffs- someone not brave, or pure, or intelligent, but is, like, really good at pottery would go in just the same as that one weird kid that is only good at music.

Hufflepuff should be an enormous umbrella of students rejected from other houses, and schools, but we can see that is not the case:

Each table has the new students from each house. Each number is roughly equal, which seems wrong. If so many students can become Hufflepuffs, why aren't there more of them?

Where are all the Hufflepuffs?

marked as duplicate by Buzz, Blackwood, SteveED, Ward, Aegon Nov 8 at 6:59

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    Related if not dupe: Do Hogwarts Houses have quotas? – Jenayah Nov 1 at 19:23
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    If there can just happen be an equal amount of students with purity, bravery, and intelligence, why can’t there also happen to be an equal amount with none of those qualities? – Alex Nov 1 at 19:25
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    Not a dupe. Tangential at best. Any connection would need enough explanation to justify an answer here. – amflare Nov 1 at 20:24
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    I'd say the other question strengthens this question. Of the four answers there, one argues that the Sorting Hat will always sort properly regardless of numbers, and the other three argue that it makes sense that the qualities would be evenly distributed among the students. This question, however, asks why you would assume equal distribution of qualities, when one of the choices is actually the sum of all qualities except for three. If the Sorting Hat sorts properly then you would expect there to be many more Hufflepuffs even if you can assume that specific qualities are evenly distributed. – Alex Nov 1 at 22:09
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    As for Slytherin reflecting "purity": Riddle/Voldemort was a half-blood, and he was Sorted into Slytherin. It was his ambition and cunning that placed him there, not his blood purity. – Wallnut Nov 2 at 9:14

Several points:

As I argued in this answer, the Sorting Hat may not have adhered to the founders' method of selection with perfect imitation. It is possible that while Hufflepuff herself would take anyone, the Sorting Hat would not quite sort that way, especially given that the Sorting Hat has control over the other three houses as well, whereas Hufflepuff herself did not. In other words, Hufflepuff herself didn't specifically value students with no qualities. She did value certain qualities but simply was willing to teach anyone because none of the other founders would take those students. But once the Sorting Hat is in complete control, it can distribute the students with no qualities across the various houses.

We know that there are specific qualities that are valued by Hufflepuff House. In the Sorting Hat's song in Philosopher's Stone we are told:

You might belong in Hufflepuff,
Where they are just and loyal,
Those patient Hufflepuffs are true
And unafraid of toil;

In the Sorting Hat's song in Goblet of Fire we are told:

For Hufflepuff, hard workers were
Most worthy of admission;

So if a student is not just, loyal, patient, true, unafraid of toil, or hard-working, the Sorting Hat might not sort them into Hufflepuff even if the student is also not brave, smart, or pure.

As Dumbledore noted at the end of Goblet of Fire, Cedric Diggory exemplified the qualities of Hufflepuff House:

"Cedric was a person who exemplified many of the qualities that distinguish Hufflepuff house," Dumbledore continued. "He was a good and loyal friend, a hard worker, he valued fair play.

If Hufflepuff was just the house for rejects, those qualities wouldn't be distinguished.

Additionally, we know that the Sorting Hat takes the student's opinion into account. As Harry tells his son:

But if it matters to you, you'll be able to choose Gryffindor over Slytherin. The Sorting Hat takes your choice into account."

"Really?"

"It did for me," said Harry.

We also know that some people did not want to be in Hufflepuff precisely because of its reputation. The first time Harry meets Malfoy, Malfoy says:

imagine being in Hufflepuff, I think I'd leave wouldn't you?"

And Harry and Hagrid later have the following conversation:

"And what are Hufflepuff and Slytherin?"

"School Houses. There's four. Everyone says Hufflepuff are a lot o' duffers, but —"

"I bet I'm in Hufflepuff," Harry said gloomily.

Hermione also implies that she wouldn't want to be in Hufflepuff:

I've been asking around, and I hope I'm in Gryffindor, it sounds by far the best; I hear Dumbledore himself was in it, but I suppose Ravenclaw wouldn't be too bad....

Therefore, it is entirely reasonable that students would specifically ask not to be sorted into Hufflepuff (whether explicitly, or by simply thinking that they don't want to be).

Thus, even if you would expect more than 25% of the student population to not be brave, smart, or pure, it is still possible to explain why Hufflepuff House does not contain more than 25% of the school.

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    You're not sorted based on what qualities you have, but on what qualities you value. See Peter Pettigrew etc. etc. – OrangeDog Nov 2 at 13:19
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    @OrangeDog I do not think it is exactly the case. You at least need to have potential to develop those qualities if not the qualities as such. Sorting Hat put people in a House that can be the best environment for their development. If you are stupid you won't end up in Ravenclaw however you'd like to look smart. So it's both - your qualities and your values. Otherwise the sorting wouldn't make sense and people would just pick their own houses. – Shana Tar Nov 4 at 5:40

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