Uses of Geometric Mean
If the quantities required for the average are drawn from the situations where the exponential law of growth (or decline) is used, then Geometric Mean would be an appropriate measure to use. For example, growth of population, deposits in a bank account attracting compound interest, depreciation charged using diminishing balance method, etc. Geometric Mean is used in these cases to determine the average percentage rates of increase or decrease from the given yearly change or other periodic change. It is also used to average the ratios.
For a better understanding of the concept, let’s take the formula of compound interest in which a sum P carries an interest rate (r) per time period, and becomes an amount (A) in n time periods:
Now, by algebraically manipulating the above formula, we can get the value of r as:
This formula can be used to calculate the percentage rate of interest per unit of time, where initial value (P), end value (A), and number of time periods (n) are given.
Besides, this relationship can be used in both conditions; when the interest rate (r) is uniform, or when the interest rate varies. For example, suppose a three-year deposit in a bank attracts r1, r2, and r3 percent per annum interest rate in the years 1, 2, and 3 respectively. Then, the average interest rate (r) can be determined as:
Given: and n = 3
Therefore,
r = [(100 + r1)(100 + r2)(100 + r3)]1/3 – 100
Alternatively,
r = GM[(100 + r1, 100 + r2, 100 + r3)]-100
Therefore, we can easily determine the average rate of interest with the help of Geometric Mean.
Example:
The percentage increase in the net worth of a company over the last 6 years is given below. Determine the average percentage increase in the net worth of the company over the 6-year period.
Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
% Increase | 5.0 | 3.4 | 7.5 | 4.8 | 2.6 | 1.8 |
Solution:
By using Geometric Mean, we can calculate the average percentage increase in the net worth of the company over the 6-year period.
Year | % Increase | Year-end Value | log X |
|---|---|---|---|
2018 | 5.0 | 105.0 | 2.0211 |
2019 | 3.4 | 103.4 | 2.0145 |
2020 | 7.5 | 107.5 | 2.0314 |
2021 | 4.8 | 104.8 | 2.0203 |
2022 | 2.6 | 102.6 | 2.0111 |
2023 | 1.8 | 101.8 | 2.0077 |
|
| Total | 12.1061 |
G.M. =
G.M. = Antilog 2.01768
G.M. = 104.15
Now,
Average percentage increase in net worth = 104.15 – 100
Average percentage increase in net worth = 4.15 per annum
Geometric Mean in Business Statistics | Concept, Properties and Uses
Different measures of central tendency are available to locate the centre of a data set. These include Arithmetic Mean, Median, Mode, Geometric Mean, and Harmonic Mean. Each of these measures is unique in its own way and has some characteristics.
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