Definition:Geometric Mean
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This page is about Geometric Mean in the context of Algebra. For other uses, see Mean.
Definition
Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers.
The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:
- $\displaystyle G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$
That is, to find out the geometric mean of a set of $n$ numbers, multiply them together and take the $n$th root.
Mean Proportional
In the language of Euclid, the geometric mean of two magnitudes is called the mean proportional.
Thus the mean proportional of $a$ and $b$ is defined as that magnitude $c$ such that:
- $a : c = c : b$
where $a : c$ denotes the ratio between $a$ and $c$.
Also see
- Results about geometric mean can be found here.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3.1.12$: Geometric Mean of $n$ quantities $G$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 3$: Natural Numbers: $\S 3.10$: Example
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: geometric mean
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: geometric mean
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: geometric mean
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: geometric mean
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: mean