Geometric Mean Properties, Applications and Limitations

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, … , x_n is given as

G.M. = (x₁. x₂ … x_n)^1⁄n

or, G. M. = (π _{i = 1}ⁿ x_i) ^1⁄n = ⁿ√( x₁, x₂, … , 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