Geometric Mean is defined as the n^{th} root of the product of numbers to be averaged. The geometric mean of numbers X_{1}, X_{2}, X_{3}.....X_{n} is given as
G = (X_{1} x X_{2} x X_{3} ..... X_{n})^{1/n}
^{The peculiar nature of growth over time is on account of compounding. For example, sales of a company were Rs.10 crore in 199798. In 19992000 they grew by 10% to Rs.11 crore. In 20002001 they grew by 20% to Rs.13.2 crore. The 20% growth rate applies to Rs.11 crore which includes the Rs.1 crore growth of the previous year. This is what is meant by compounding.}
PROPERTY
The product of the quantity ratios will remain unchanged when the value of geometric mean is substituted for each individual value. This may be seen by substituting 0% for 100% and 50% in the above example.
USES
The geometric mean is used to find the average percent increase in sales, production, population or other economic or business series overtime.
Example 8
The following data relates to Voltas Ltd.
Year

Sales Rs. in millions

19981999

6670.0

19992000

7794.6

20002001

9176.2

The growth rate for the year 19992000 
= 

x 100 
= 16.86% 
The growth rate for the year 20002001 
= 

x 100 
= 17.73% 
We can find that the sales of Voltas Ltd. has been increasing year by year, but at different growth rates. Now, the compounded annual growth rate can be arrived at by taking the geometric mean for the two quantity ratios.
G.M.

= 

= 
1.1729 
Growth rate = 1.1729  1 = 0.1729 or 17.29%
Thus, the compounded annual sales growth rate of Voltas Ltd. for the years 199899, 19992000 and 20002001 is 17.29%.
Geometric mean is most frequently used in finding out compound interest. The expression used in compound interest formula is
P_{n} = P_{0}(1 + r)^{n}
where,
P_{n} = The value at the end of period n
P_{0} = The value at the beginning of the period
r = Rate of compound interest per annum (expressed as a fraction)
^{ n = Number of years. }