Geometric Mean

Consider a list of \( n \) values \( X = \{x_1, x_2, \dots, x_n\} \). The geometric mean is defined by the nth root ( \( \sqrt[n]{} \) ) of the product of the values.

$$ \bar{x}_{geom} = \sqrt[n]{\prod_{i=1}^n{x_i}} $$

Example: The list of \( 3 \) numbers \( \{1, 10, 100 \} \) has for geometric mean \( \sqrt[3]{1 \times 10 \times 100} = 10 \), whereas it has for arithmetic mean \( 55.5 \).

To get a geometric representation, if we take the geometric mean of the sides of a rectangle, then we find a value \( c \) which could be the length of one side of a square of area identical to the original rectangle.

Example: Consider a rectangle of \( 6 \times 10 \), with an area of \( 60 \). The geometric mean of \( 6 \) and \( 10 \) is \( \approx 7.746 \). And a square of side length \( 7.746 \) has an area of \( \approx 60 \).

When the values are assigned coefficients, it is called a weighted geometric mean.

Using the mathematical formula : //Python
import numpy as np
def geometric_mean(iterable):
a = np.array(iterable)
return a.prod()**(1.0/len(a))
or to avoid a potential number overflow ://Python
import numpy as np
def geometric_mean(iterable):
a = np.log(iterable)
return np.exp(a.sum()/len(a))