# Geometric Mean: Characteristics, Applications and Limitations

8th February 2020A 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 x_{1}, x_{2}, … , x_{n} is given as

G.M. = (x_{1}. x_{2} … x_{n})^{1⁄n}

or, G. M. = (π _{i = 1}^{n} x_{i}) ^{1⁄n }= ^{n}√( x_{1}, x_{2}, … , x_{n}).

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

**Advantages of Geometric Mean**

- A geometric mean is based upon all the observations
- It is rigidly defined
- The fluctuations of the observations do not affect the geometric mean
- It gives more weight to small items

**Disadvantages of Geometric Mean**

- A geometric mean is not easily understandable by a non-mathematical person
- If any of the observations is zero, the geometric mean becomes zero
- If any of the observation is negative, the geometric mean becomes imaginary