I imagine there is a trend where, in higher dimensions, coordinates have a greater tendency to be near zero?

In a 0-sphere, x is either -1 or +1. In a 1-sphere, each of x, y, is (informally speaking) more likely to be near +/- 1 than near 0. A 2-sphere gives us uniform distribution for each coordinate. So I suppose that the coordinates of a 3-sphere are more likely to be near 0 than near +/- 1, and this tendency is more pronounced, the higher the dimension gets. (?)

n-spheres are funny things. Intuition about them is often misleading. (See, for example, the comments expressing skepticism about my original uniform-distribution observation.)

----

EDIT. Ooo, some interesting questions here. What is the limiting behavior of the distribution of x on the n-sphere, as n -> +infinity? I imagine the graph looks like a narrower & narrower spike at x = 0.

Now, suppose we scale the graph of the distribution horizontally by some appropriate function of n. That is, let f_n be the probability distribution of x on the n-sphere. Is there some function g:Z -> R so that the function x -> f_n(x / g(n)) has a limit as n -> +infinity? Does it approach (wild guess) a normal distribution? If so, is this fact (even wilder guess) a special case of some kind of central-limit-theorem-ish statement that holds for geometrical objects?