## Why Are Average People Overweight?

Bob is an average person: Bob's height equals the average height in the population, and Bob's weight equals the average weight in the population.

So why is Bob considered to be overweight?

Notes
• This is a puzzle, not a joke.
• "Overweight" is meant relative to the population.
• It does not depend on the distribution.
• Source: Nobuyuki Yoshigahara, Puzzles 101: A Puzzlemaster's Challenge, A K Peters (2003)

The weight is roughly proportional to the volume, which is roughly proportional to the third power of the height (a better approximation may in fact be the second power of the height, which works just as well -- see the Note below).

Example:

• Take 3 people with heights 1, 2, 3, and corresponding weights 1 ( = 13), 8 ( = 23), 27 ( = 33).
• Average height = (1 + 2 + 3) / 3 = 2.
• Average weight = (1 + 8 + 27) / 3 = 12.
• So Bob is overweight: people of his height (2) weigh (on the average) 8, not 12.
A larger example that includes Bob (9 people, Bob is the 6th person):

 Height 1 1 1 2 2 2 3 3 3 average height = 2 Weight 1 1 1 4 8 12 20 27 34 average weight = 12

Note:

Let f(h) = E[W | H = h] be the average weight of all people of height h. If f(h) is a strictly convex function of h (for example: f(h) = c ha for some a > 1 and c > 0), then the above phenomenon always holds (regardless of the distribution):

E[W] = E[E[W | H]] = E[f(H)] > f(E[H]).

(W = weight, H = height, E = expectation)

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