If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.

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