[Tex]GM_w=Antilog\frac{\sum{W~logX}}{\sum{W}} [/Tex]

= [Tex]Antilog[\frac{15.896}{20}] [/Tex]

= Antilog 0.7948

G.M. = 6.23

Note: It should be kept in mind that if any value of the given data set is negative or zero, then the value of Geometric Mean cannot be calculated. Also, if the given data set is open-ended distribution, the value of Geometric Mean cannot be obtained because to determine G.M., the X values of all class intervals are required.

Properties of Geometric Mean

Different algebraic properties of Geometric Mean are as follows:

1. The product of n values is the same as the product obtained by replacing each value with its geometric mean. Simply put,

X₁ x X₂ x X₃ x X₄ x ……….. x X_n = (GM)ⁿ

For example, the geometric mean of 4, 8, 4, and 2 is equal to [Tex]\sqrt[4]{4\times{8}\times{4}\times{2}}=4 [/Tex]

Also, the product of the values; 4, 8, 4, and 2 is 256 which is equal to 4⁴.

2. The product of the proportions of values lesser than or equal to the geometric mean to the geometric mean is equal to the product of the proportions of the geometric mean to the values greater than the geometric mean. Simply put,

[Tex]\frac{GM}{X_1(\leq{GM})}\times{\frac{GM}{X_2(\leq{GM})}\times}……\times{\frac{GM}{X_r(\leq{GM})}}=\frac{X_{r+1}(>GM)}{GM}\times{\frac{X_{r+2}(>GM)}{GM}\times……\times{\frac{X_n(>GM)}{GM}}} [/Tex]

Taking the previous example into consideration, the values less than or equal to the geometric mean are 4, 4, and 2; and the value greater than the geometric mean is 8. Now,

[Tex]\frac{4}{4}=\frac{4}{4}=\frac{4}{2}=\frac{8}{4}=2 [/Tex]

3. If the number of observations and geometric mean of two or more series are given, then their geometric mean can be determined. The formula of calculating combined geometric mean of two series is as follows:

Combined Geometric Mean = [Tex]Antilog\frac{n_1\times{log~GM_1+n_2\times{log~GM_2}}}{n_1+n_2} [/Tex]

For example, the geometric mean of a set of 10 observations is 18.5 and the geometric mean of another set of 15 observations is 7.8. The combined geometric mean of both the sets will be,

Combined Geometric Mean = [Tex]Antilog\frac{n_1\times{log~GM_1+n_2\times{log~GM_2}}}{n_1+n_2} [/Tex]

= [Tex]Antilog\frac{(10\times{log~18.5})+(15\times{log~7.8})}{10+15} [/Tex]

= [Tex]Antilog\frac{(10\times{1.26})+(15\times{0.892})}{25} [/Tex]

= Antilog 1.0392

Combined Geometric Mean = 10.94

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:

[Tex]A=P(1+\frac{r}{100})^n [/Tex]

Now, by algebraically manipulating the above formula, we can get the value of r as:

[Tex]r=[\sqrt[n]{A/P}]\times{100} [/Tex]

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: [Tex]A=P(1+\frac{r_1}{100})(1+\frac{r_2}{100})(1+\frac{r_3}{100})   [/Tex] and n = 3

Therefore, 

[Tex]r=[\sqrt[3]{\frac{P(\frac{100+r_1}{100})(\frac{100+r_2}{100})(\frac{100+r_3}{100})}{P}}-1]\times{100} [/Tex]

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.