I've always preferred "normal" to "average," and I have always considered the "normal range" to run, roughly, from the 20th to the 85th percentile.
It is tempting to use test scores to compare, because numbers are easily compared and there is no grey area about whether 60 is better than 40 — it simply is. As a PP notes, school admission offices are generally very savvy about weighing the non-numerical parts of an application. If a school works well with "average" students, and most schools do, then "average" students should do fine there.
Peter
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Disclaimer: The anonymity here makes me uncomfortable; it's easy to be uninformed, personal, or simply mean-spirited if people don't identify themselves. For that reason, I have an account so you know whose words you're reading. I have more than 20 years' experience as a teacher and administrator in independent schools, and I have counseled hundreds of students in finding their next schools. I hope I can be helpful to some folks. If you don't like something I've said, you're in good company — there's a long line of past students and parents ahead of you.

If you want to chat further, please feel free to contact me offline: peter <at> arcpd <dot> com