Calculation
The geometric mean of a data set is given by:
The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between.
The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined:
and
then an and hn will converge to the geometric mean of x and y.
This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano–Weierstrass theorem) and the fact that geometric mean is preserved:
Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
Read more about this topic: Geometric Mean
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