At least some content in this article is derived from information featured in Harry Potter: Hogwarts Mystery. Spoilers will be present within the article.
"While you are at Hogwarts, your triumphs will earn your House points, while any rule-breaking will lose House points. At the end of the year, the House with the most points is awarded the House Cup, a great honour."
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Exactly how points are tabulated and recorded is not explained in detail. It is implied in several books that simply stating the addition or subtraction of points magically adjusts the score accordingly. It's not certain if this occurs in all cases, such as when Dumbledore adjusts the scores at the end of Harry Potter and the Philosopher's Stone. The four giant hourglasses, however, seemingly adjusts their contents accordingly to the house points granted or retracted by the teachers' wordings.
In Harry Potter and the Half-Blood Prince, Severus Snape takes 70 points from Gryffindor before term starts, saying that Gryffindor would have negative 70 points, but in Harry Potter and the Order of the Phoenix, Snape was unable to take points from Gryffindor when they had no points left. It is possible, however, that Snape's next words after "In that case, Potter, we will simply have to..." would have been to send to Gryffindor into negative points, but he was unable to get them out before being interrupted by Professor McGonagall. It can also be assumed he was being sarcastic or joking cruelly about the situation. Also, in Harry Potter and the Chamber of Secrets, Minerva did not take any points from Gryffindor when Harry and Ron took the Flying Ford Anglia, since it was before term started, which shows Snape's deduction of 70 points 4 years later as unorthodox.
In the films and video games from Harry Potter and the Order of the Phoenix, the house points are earned and counted in the Great Hall to the right of the staff table. However in the first video games, they are also counted in the Great Hall, despite the hourglasses being located in the Entrance Hall.
In Harry Potter and the Philosopher's Stone, Severus Snape deducts one point from Gryffindor for Harry's statement that Hermione could answer his questions and suggestion that he ask her. This, and the point that he takes for Harry's supposedly setting Neville Longbottom up for failure, are the only instances shown in the books of a single point being deducted for a transgression. In the extended cut of the Film, this is a straight 5 points. In the original version of the Film, it's omitted entirely. They are normally deducted at either 5 or 10 points for small infractions, and multiples of these numbers for greater ones, though in the earliest Harry Potter video game, certain individuals, such as a particular Hufflepuff prefect, deducted them in unusual amounts, such as 6.
In the films, the role of house points is almost exclusively limited to the first film. The other 7 films make almost no mention of them outside of Pomona Sprout awarding Hermione 10 points for identifying both the properties and dangers of the Mandrake Root (as opposed to 20 for both in the books) in Harry Potter and the Chamber of Secrets, Professor Snape deducting 5 points from Hermione being an "insufferable know-it-all" during Harry Potter and the Prisoner of Azkaban and a deleted scene of Harry Potter and the Goblet of Fire during the Yule Ball when Snape deducts 10 points each from Fawcett and Stebbins when they were caught in a Hogwarts carriage. The films make no mention of the House Cup or point totals is mentioned in any film outside of Harry Potter and the Philosopher's Stone.
In the GBA version, Professor McGonagall, instead of awarding 10 points to Harry and Ron and deducing 5 from Hermione, awards 50 points to Gryffindor for saving Hermione.
