If there are only k numbers x(1),x(2)....,x(k), the geometric mean is the kth root of the product of these k numbers.
Example: find the geometric mean of 4,3,7,8
We want the fourth root of 4 x 3 x 7 x 8 = 672
=(672)^(1/4) = 5.09146 is the geometric mean.
The geometric mean is normally defined only for a set of positive numbers.
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The difference between arithmetic and geometric mean you can find in the following link: "Calculation of the geometric mean of two numbers".
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Geometric mean of 6 and 20 = sqrt(6*20) = sqrt(120) = 10.95445
To find a geometric mean, we multiply all of the terms together and take the nth root of the result (where n is the number of terms we are averaging). With 10 and 6, we find the geometric mean is the square root of 10*6 = 60. Sqrt(60) = 2*sqrt(15).