Geometric mean

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 (as opposed to the arithmetic mean which uses their sum). 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






(


∏

i
=
1


n



x

i



)



1
n



=




x

1



x

2


⋯

x

n




n





{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}
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 (1/8), which is 1/2, that is,






4
⋅
1
⋅
1

/

32


3



=
1

/

2


{\displaystyle {\sqrt[{3}]{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.
The geometric mean can be understood in terms of geometry. The geometric mean of two numbers,



a


{\displaystyle a}
and



b


{\displaystyle b}
, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths



a


{\displaystyle a}
and



b


{\displaystyle b}
. Similarly, the geometric mean of three numbers,



a


{\displaystyle a}
,



b


{\displaystyle b}
, and



c


{\displaystyle c}
, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers.
The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)

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