Geometric Mean

     The geometric mean is the mean or average which indicates the central tendency or typical value of a set of numbers. This is similar to arithmetic mean, which is the sum of adding the set of numbers and then dividing the sum of count number in set, n, and the number are multiplied and the nth root of resulting value is taken. The geometric mean only applies to positive numbers.

Formula Of Geometric Mean

Geometric Mean = ((X₁)(X₂)(X₃)........(X_N))^1/N
Where
              X = Individual score
              N = Sample size (Number of scores)
Geometric mean is also used for set of numbers whose values are multiplied together or exponential in nature, such as growth of growth of human population or interest rates of a financial investment. The geometric mean is one of the three classic Pythagorean means, aforementioned arithmetic mean and harmonic mean.

Calculation of Geometric Mean

 The geometric mean of data set [ a₁, a₂, --------, a_n ] is given by
^{$\left(\prod_{i = 1}^{n} a_{i}\right)^{1/n}$} = $ \sqrt[n]{a_{1}a_{2}......a_{n}}$
The geometric mean of a given set is less than or equal to the given set of arithmetic mean (the two mean of equal when all the members of a data set are equal) this allows the definition for arithmetic geometric mean the two values always lie in between the geometric mean is also arithmetic harmonic mean in that if two sequences (a_n) and (h_n) are defined:
a_{n + 1} =  $\frac{an + hn}{2}$ , a_o   = x and
h_{n + 1} = $\frac{2}{1/an + 1/bn}$, h_o = y.
Then a_n is converge to the geometric mean of x. h_n is converge to the geometric mean of y.
The sequence for converge to a common limit ( which can be shown by Bolzano - Weierstrass theorem) the geometric mean is given by
$\sqrt{a_{i}h_{i}}$  = $\sqrt{\frac{a_{i} + h{i}}{1/a_{i} + 1/h_{i}}}$  =  $ \sqrt{a_{i + 1}h_{i+1}}$

Exponential Growth of Geometric Mean

The geometric mean is more approximate than the arithmetic mean for explaining exponential growth.