# Geometric Mean: Characteristics, Applications and Limitations

8th February 2020

A geometric mean is a mean or average which shows the central tendency of a set of numbers by using the product of their values. For a set of n observations, a geometric mean is the nth root of their product. The geometric mean G.M., for a set of numbers x1, x2, … , xn is given as

G.M. = (x1. x2 … xn)1⁄n

or, G. M. = (π i = 1n xi1⁄n n√( x1, x2, … , xn).

The geometric mean of two numbers, say x, and y is the square root of their product x×y. For three numbers, it will be the cube root of their products i.e., (x y z) 1⁄3.

#### Properties of Geometric Means

• The logarithm of geometric mean is the arithmetic mean of the logarithms of given values
• If all the observations assumed by a variable are constants, say K >0, then the G.M. of the observation is also K
• The geometric mean of the ratio of two variables is the ratio of the geometric means of the two variables
• The geometric mean of the product of two variables is the product of their geometric means