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In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1, x2,..., xn, the geometric mean is defined as
or, equivalently, as the arithmetic mean in logscale:
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, 2 ⋅ 8 = 4 {\displaystyle {\sqrt {2\cdot 8}}=4} . As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product , which is 1/2, that is, 4 ⋅ 1 ⋅ 1 / 32 3 = 1 / 2 {\displaystyle {\sqrt{4\cdot 1\cdot 1/32}}=1/2} . The geometric mean applies only to positive numbers.
The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time. It also applies to benchmarking, where it is particularly useful for computing means of speedup ratios: since the mean of 0.5x and 2x will be 1.