How do you calculate the geometric mean?

Answer 1

The geometric mean, by definition, is the nth root of the product of the n units in a data set.

For example, the geometric mean of 5, 7, 2, 1 is (5x7x2x1)1/4 = 2.893

Alternatively, if you log transform each of the individual units the geometric will be the exponential of the arithmetic mean of these log-transformed values.

So, reusing the example above, exp[ ( ln(5)+ln(7)+ln(2)+ln(1) ) / 4 ] = 2.893

I agree with the above BUT... with two numbers, a geometric mean is the square root of the product (result of multiplication) of the two numbers while the arithmetic mean is half of the sum of the two numbers. For example, 2 and 8. The geometric mean is 2*8 = 16, sqrt(16) = 4. The arithmetic mean is (2+8)/2 = 5. Both give a number somewhere between those that contribute AND when the numbers are the same, both will agree. For 6 and 6, sqrt(36) = 6, 12/2 = 6. In other cases, they have different properties which make them advantageous in different places. BUT you should think of the two has having the same goal and being very similar in form (just turning addition to multiplication and multiplication to exponentiation).

The earlier answer is better because it shows the generalization beyond two numbers (that is, when you have four numbers, you will multiply them all together and take the 4th root, rather than the square root), I added this to draw the similarity to a conventional mean and give an example that you could follow in your head.

(Just wondering, why is it useful to show that for exponents multiplication becomes addition? -- I am not following the relevance to a geometric mean.)