Multiple times in Harry Potter, we see house points being awarded in multiples of 5, but when playing the games, I've come across some odd numbering for house points. (394, etc.)
Can house points be awarded in non-multiples of 5?
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For most students, points were given or taken one at a time.
It is only in unusual cases where more than a single point was awarded (or taken). We don't see this much, though, because the books only feature unusual situations.
If you consider the first book, where the Trio is in their first year and Hogwarts is at its most 'normal', you tend to see points given in very small increments (especially compared to later books). The only exception is the end of term feast, where Dumbledore skews everything wildly by giving out 170 points at once to four people.
Given that those 170 points were enough to completely change the scores and take a very low score into the winning position, we can assume that the typical change in points across a typical day or week must be very low.
In general, I'd be willing to bet that most houses have a net change of only a few points per day (up or down).
If you are interested in the spread of points given in the books (which does really show just how skewed a view we get) you can check out this page on Pottermore, as shared by ibid in the comments above.
A single point was taken twice (in the same class) from Harry by Professor Snape in Harry Potter and the Sorcerer's Stone.
And a point will be taken from Gryffindor House for your cheek, Potter.
Later in the class, he says this line:
You—Potter—why didn't you tell him not to add the quills? Thought he'd make you look good if he got it wrong, did you? That's another point you've lost for Gryffindor.
Thus, it seems likely that it was not unusual for points to be given or recieved in 1-point increments or decrements.
Pottermore has a semi-detailed analysis of the house points awarded and deducted through out the whole series.
In Philosopher's Stone it says that the total number of points deducted was 187.
This would mean that at some point at least 2 points were added or 3 points were deducted in some fashion to deviate from 5-point system. Granted this could be as individual points (i.e. +/- 1) or as some random number (i.e. +17 or -23).
The answer is simple. Here's something Dumbledore says at the end of book 1:
"Now, as I understand it, the house cup here needs awarding, and the points stand thus: In fourth place, Gryffindor, with three hundred and twelve points; in third, Hufflepuff, with three hundred and fifty-two; Ravenclaw has four hundred and twenty-six and Slytherin, four hundred and seventy- two."
There's no possible way for that to happen only from multiples of 5. Therefore, it must be that points are added in "awkward numbers."