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I am aware that the geometric mean is often used with the lognormal distribution, because then it directly relates to the arithmetic mean with the normal distribution.

But I was trying to think of an intuitive defintion of the geometric mean.

For example, the median can be explained as "the data point such that half of the data points have higher values and the other half have lower values."

Is there a similar definition for the geometric mean?

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    $\begingroup$ The mean is not what you described... $\endgroup$ – Don Thousand Jan 21 '20 at 23:45
  • $\begingroup$ @DonThousand the mean is harder to describe without a graph I think, but there is still an intuitive explanation because people understand the idea of averages $\endgroup$ – Jess Jan 21 '20 at 23:47
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    $\begingroup$ @Jess:sadly, I don't think many people do understand the idea of averages, but most of them pretend they do, so you can get away with explaining the arithmetic mean in terms of averages. As regards the geometric mean, I wish you well in your search for a good informal explanation. $\endgroup$ – Rob Arthan Jan 22 '20 at 0:01
  • $\begingroup$ Not sure if this will help, but there are some applications of the geometric mean listed on the Wikipedia page. $\endgroup$ – angryavian Jan 22 '20 at 0:06
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    $\begingroup$ @Crostul: that's interesting. What was the context? $\endgroup$ – Rob Arthan Jan 22 '20 at 0:27
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The geometric mean of two nonnegative numbers $a$ and $b$ can be understood as the side length of a square with area equal to a rectangle with side lengths $a$ and $b$. For three numbers, it’s the side length of a cube with equal volume to a box with side lengths given by the three numbers, and so on.

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This is of course not a 'non-mathematical definition', but I hope it is quite intuitive...

A geometric mean (see Wikipedia) of values $A$ and $B$ is a value $Q=\sqrt{A\cdot B}$, such that a square with side $Q$ has an area equal to that of a rectangle $A$ by $B$.

For more variables, a geometric mean of values $x_1, x_2, \ldots x_n$ is $$\sqrt[n]{x_1\cdot x_2 \cdot \ldots \cdot x_n}$$ which is a length of the edge of an $n$–dimensional hypercube, which has the same (hyper)volume as a (hyper)cuboid with edges' lengths $x_1, x_2, \ldots x_n$.

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In music theory, where an octave represents a doubling of sound frequency, there is a correspondence to arithmetic, harmonic, and geometrical means: the arithmetic mean (mathematically, of sound frequencies) between C and C one octave apart is the 'perfect fifth'; the harmonic mean is the 'perfect fourth', and the geometric mean, the so-called 'tritone'. I'm actually a little rusty on my music theory, but the fifth is simply the fifth note in do-re-me-fa-sol-la-ti, e.g. sol; fa being the fourth. The tritone is, as the mathematics would imply, between the fourth and the fifth, but does not correspond to one of the whole notes. Nevertheless, musically it does exist as a reproducible entity in certain chords.

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  • $\begingroup$ Wow that's really cool! $\endgroup$ – Jess Jan 22 '20 at 1:46

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