On the Harry Potter Wiki page for Hogwarts, it has the following to say about first years:
First years are typically eleven to twelve years of age.
Simple question then: is it ever explained in-universe or by J K Rowling why the age of admission is 11-12 rather than perhaps a slightly younger or older age? Is there any specific importance of the age 11 when it comes to magical development of skills, or it is simply that it corresponds with the standard muggle schooling system?
It's to do with the British school system.
Most school tracks move from primary (or infant/junior school) covering from ages 5 to 11, and then onto secondary (senior or high school) for ages 11-16 (sometimes up to 18). There is another track of primary (5-9), middle (9-13), and secondary (13-16+).
Hogwarts is modeled on a typical high school with boarding, so it makes sense that admission would start at 11 years. For one thing, I don't think there have been all that many boarding schools for primary school ages (at least in more recent times), and also the difference between a 5 year old and 11 year old is quite a bit, and it was probably easier to write 11 year old characters getting up to the antics they do in the book.
TL;DR Hogwarts is set in a UK system, and so assumes a two stage school process with a transition between primary and secondary schools at the age of 11.
As many have stated above, the UK school system typically comprises two 7-year cycles: primary (4 to 11 years) and secondary (11 to 18 years). The years in primary are usually defined as P1 to P7. The years in secondary are usually defined as numbered year up to 6th, and then as lower 6th and upper 6th for the final two years. Sometimes the final two years are in separate institutions called Sixth Form Colleges.
In the past the transition between the two systems, primary and secondary, was governed by a famous examination called the 11+ (emphasing the importance of this age of transition). Those performing well in the academic styled exams were admitted to Grammar Schools, with an academic ethos, and those performing less well (contested, see below) to Secondary Modern schools with more of a vocational/trades focus. This system was in place from the end of WWII onwards after the Butler Education Act 1944, that made secondary as well as primary education free for all students.
Although this system was very successful by many metrics, the 1960s saw a set of changes prompted by the belief that it was very unfair to stream students at such a young age of 11, and in fact created large barriers to social mobility; analysis showed that the class of the parents, rather than than the ability of the child was the main factor in success in the 11+, often linked to how much the parents could support their own children through the system. Therefore, in most of the UK the 11+ is no longer used, and the secondary system now has Comprehensive Schools that try and cover all these types of education in a single school.
There remains a big debate about the number of comparatively wealthy people who by-pass this more egalitarian system, and send their children to fee-paying schools, which in the UK are counter-intuitively called "public" schools. Arguably Hogwarts is most like one of these elite private schools, like Eton, Harrow and Westminster, which are often centred on a boarding culture. J.K. Rowling herself went to a Comprehensive School, Wyedean School and College, near the border between Wales and England.
In the broader context, there is a long tradition of boarding school literature in British children's literature: school story. This reached its peak in the late 19th and early 20th centuries. So the Harry Potter stories can be seen as a revival of this tradition, albeit with a co-educational (boys and girls) rather than a single sex school, as had been the norm in the tradition.
Obscuri almost always die by age 10. Perhaps the in-universe reason to start at age 11 was to prevent a potential Obscurus coming to Hogwarts and wreaking havoc. Especially since they took in Muggle children (against the wishes of Slytherin), who were more likely to be persecuted for there magic, and hence more likely to be Obscuri:
... [Hogwarts was founded in] an age when magic was feared by common people, and witches and wizards suffered much persecution.” (COS)
One of the founders, Salazar Slytherin, felt that Muggle-born students were untrustworthy. There is no evidence that he was prejudiced against them (this seems to be merely legend). All it says is that he felt they couldn't be trusted. It may be he felt they were untrustworthy due to the fact that a high percentage of them were possible Obscuri, and Slytherin suspected one may survive past age 10 (as we see is indeed possible, with Credence.)
In fact, that would explain why Slytherin built the Chamber of Secrets with a Basilisk inside: Not as an insane plot so some evil descendant of his would be able to murder innocent kids in the castle, but as backup plan to kill an Obscurus should one terrorize the school. There is no evidence for this, but considering that Slytherin was a great wizard and he and Gryffindor were once friends, that seems to me to be far more reasonable than the common assumption that Slytherin was insane and unbalanced (which he would have to be if he built the Chamber of Secrets for the reason the legend states.)
Answer: The age 11 is based on the "in-universe" detail of sleeping at the school. The students are only accepted to Hogwarts after they complete primary school, because they sleep in the dormitories, which is a level of independence, which is after completing primary school, and not specifically the age number 11. for more information http://harrypotter.wikia.com/wiki/Gryffindor_Boys%27_Dormitory Although the author said she uses numerology, and in numerology double-digits such as 11 [a double one] are special, the fact that primary schools are until age 11 is the reason which is so obvious, that it was not stated. Also in-universe, we find that 11 was the age Harry planned to switch schools, a fact which supports the above idea.
It's explained that Hogwarts has a automatic magical record of children born with magical ability that are eligible for study at Hogwarts. Mcgonagall checks this record once a year to send acceptance letters to those children approaching their 11th year of age (so they students start between ages 10-11).
The specific chosen year in-universe has to do with JRK's use of numerology. 11 is a magical number not to mention 11 is around the time that kids are able to start having some conscious control over their magical abilities and are allowed to buy wands. Giving them wnads before that is like handing car keys to someone who doesn't know how to drive.
source: https://www.pottermore.com/writing-by-jk-rowling/the-quill-of-acceptance-and-the-book-of-admittance
