I'll add a fast guess from some earlier, solid experiences: Let R be the set of real numbers and n a positive integer. Consider R^n, the common n-dimensional space. Assume this Rn is Euclidean, that is, has the usual inner product, norm, metric, topology, and measure (area, volume, e.g., in the sense of classic Lebesgue measure* theory). With this overly detailed description of context (WHEW!), we move on to the main point:
In the R^n, consider a sphere. To be more clear, for some positive real number r, consider the set of all points in R^n with distance <= r from the the origin, that is, the point in R^n with zeros in all its coordinates.
Now for the big point: Consider the volume of that sphere and let n grow. Will discover that very soon nearly all the volume of the sphere is in a thin shell just inside the surface of the sphere. Or to be more explicit, for positive real number s a little less than r, nearly all the volume is in the set of points x where the distance from the origin d(x) is
s <= d(x) <= r
I encountered this in some analysis, I have forgotten, maybe generation of random points in the sphere independent and uniformly distributed in the sphere.
So, for the 10 or so body measurements of the pilots and the women, it's going to be tough (small probability for nearly any reasonable, non-pathological probability distribution on the sphere) to expect all 10 measurements to be close to the expectations, i.e., the center of the sphere (for people, the center will not be at the origin, that is, all n coordinates zero).
There is a counterexample: The girl I dated in high school, the most beautiful female I ever saw, in person or otherwise, was essentially perfect on all dimensions!
In the R^n, consider a sphere. To be more clear, for some positive real number r, consider the set of all points in R^n with distance <= r from the the origin, that is, the point in R^n with zeros in all its coordinates.
Now for the big point: Consider the volume of that sphere and let n grow. Will discover that very soon nearly all the volume of the sphere is in a thin shell just inside the surface of the sphere. Or to be more explicit, for positive real number s a little less than r, nearly all the volume is in the set of points x where the distance from the origin d(x) is
s <= d(x) <= r
I encountered this in some analysis, I have forgotten, maybe generation of random points in the sphere independent and uniformly distributed in the sphere.
So, for the 10 or so body measurements of the pilots and the women, it's going to be tough (small probability for nearly any reasonable, non-pathological probability distribution on the sphere) to expect all 10 measurements to be close to the expectations, i.e., the center of the sphere (for people, the center will not be at the origin, that is, all n coordinates zero).
There is a counterexample: The girl I dated in high school, the most beautiful female I ever saw, in person or otherwise, was essentially perfect on all dimensions!