[connoredel]

There is an analogy to clustering (an unsupervised learning technique) here.

Take the simple case of 2 dimensions (each observation is plotted in 2D space) with possible values of 0-10. Let's say the extreme (far from average) space is within 5% of the border. The total extreme area is (10x10)-(9x9) = 19 (i.e. 19%). Now add a 3rd dimension. The extreme "volume" in 3d space is now (10x10x10)-(9x9x9) = 271 (i.e. 27%). You can see where this is trending. Add enough dimensions, and every observation is now "extreme." They become so far apart that each observation almost deserves its own cluster, and you lose any idea of similarity.

Back to this particular article: when you _add_ (or average) all of the dimensions -- like you do on an exam -- suddenly they are close again.

[vlasev]

Here's another look. If you have variables X_1, ..., X_n that are independent and random from normal distributions, if you want someone to be within 1 standard deviation from the mean in EACH dimension, then you are looking at a probability of that happening equal to about 0.68^n, which becomes really small for even a moderate n.

[fats_tromino]

This is the most succinct and clearest explanation of what's going on. I see this discussed a lot when people talk about the curse of dimensionality. Another very simple example is the example of a n-hypercube with edge length 1/2 embedded in the unit n-hypercube. As n increases, the volume of the unit hypercube is constant (1), whereas the volume of the smaller hypercube is decreasing at an exponential rate.