Geometric mean

In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of ⁠ n {\displaystyle n} ⁠ numbers is the nth root of their product, i.e., for a collection of numbers a1, a2, ..., an, the geometric mean is defined as a 1 a 2 ⋯ a n t n . {\displaystyle {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}.} Hence, the geometric mean of two numbers is the square root of their product, for example with numbers ⁠ 2 {\displaystyle 2} ⁠ and ⁠ 8 {\displaystyle 8} ⁠ the geometric mean is 2 ⋅ 8 = {\displaystyle \textstyle {\sqrt {2\cdot 8}}={}} 16 = 4 {\displaystyle \textstyle {\sqrt {16}}=4} . The geometric mean of the three numbers is the cube root of their product, for example with numbers ⁠ 1 {\displaystyle 1} ⁠, ⁠ 12 {\displaystyle 12} ⁠, and ⁠ 18 {\displaystyle 18} ⁠, the geometric mean is 1 ⋅ 12 ⋅ 18 3 = {\displaystyle \textstyle {\sqrt[{3}]{1\cdot 12\cdot 18}}={}} 216 3 = 6 {\displaystyle \textstyle {\sqrt[{3}]{216}}=6} . When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm ⁠ ln {\displaystyle \ln } ⁠ of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale by using the exponential function ⁠ exp {\displaystyle \exp } ⁠, a 1 a 2 ⋯ a n t n = exp ⁡ ( ln ⁡ a 1 + ln ⁡ a 2 + ⋯ + ln ⁡ a...