leafdude replied to your post: How is it that I never thought of the averaging…
What is this?
…Oh right, I probably should have included that part, haha.
In any set, there is at least one member of the set less than or equal to the average, and at least one member of the set greater than or equal to the average.
It’s completely obvious now that I’ve read it, but it’s one of those seemingly-trivial results that has some surprising power to it, like pigeonhole principle. It’s a key part of weight shifting proofs, for example; proofs where you want to show that some object with desired properties exists by assigning weights in such a manner that objects with large (or small) weight has the properties you want, and then applying averaging principle to show that there must exist some object in the set with large or small enough weight to have those properties.