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The nice thing about the geometric mean is that it offers a good measure of central tendency for high and low extremes. Say you are trying to calculate something which involves the height of some very large cliff. You guess that the cliff is at most 100,000 feet and at the least 10,000 feet.
You could take the arithmetic mean:
( 100,000 + 10,000 ) divided by 2 = 55,000 feet
But notice that this is roughly fives times as large as your lower estimate and half of your upper estimate. Because the upper estimate is large, the arithmetic mean favors the higher extreme.
The geometric mean, however, is calculated this way:
Square root of ( upper * lower )
In this case,
sqrt( 100,000 * 10,000 ) = about 32,000 feet
This solution is roughly three times as big as your lower estimate and three times as small as your upper estimate. It has this nice property of proportionality, which can be useful and perhaps more appropriate for estimation.
When to use the geometric or the arithmetic mean is up to you. Isn’t it nice to have options?
