Geometric Mean

It is another measure of central tendency based on mathematical footing like arithmetic mean. Geometric mean can be defined in the following terms:

Geometric mean is the nth positive root of the product of “n” positive given values

Hence, geometric mean for a value X containing n values such as {x_1},{x_2},{x_3},...,{x_n} is denoted by G.M of X and given as under:

G.M{\text{ of }}X = \overline X = \sqrt[n]{{{x_1} \cdot {x_2} \cdot {x_3} \cdot \cdots \cdot {x_n}}}

(For Ungrouped Data)

If we have a series of n positive values with repeated values such as {x_1},{x_2},{x_3},...,{x_k} are repeated {f_1},{f_2},{f_3},...,{f_k} times respectively then geometric mean will becomes:

G.M{\text{ of }}X = \overline X = \sqrt[n]{{{x_1}^{{f_1}} \cdot {x_2}^{{f_2}} \cdot {x_3}^{{f_3}} \cdot \cdots \cdot {x_k}^{{f_k}}}}

(For Grouped Data)

Where n = {f_1} + {f_2} + {f_3} + \cdots + {f_k}

Example:

Find the Geometric Mean of the values 10, 5, 15, 8, 12

Solution:
Here{x_1} = 10, {x_2} = 5{x_3} = 15{x_4} = 8{x_5} = 12 and n = 5

G.M{\text{ of }}X = \overline X = \sqrt[5]{{10 \times 5 \times 15 \times 8 \times 12}}
\overline X = \sqrt[5]{{72000}} = {(72000)^{\frac{1}{5}}} = 9.36

Example:

Find the Geometric Mean of the following Data

X
13
14
15
16
17
f
2
5
13
7
3

Solution:
We may write it as given below:

Here {x_1} = 13, {x_2} = 14{x_3} = 15{x_4} = 16{x_5} = 17

{f_1} = 2, {f_2} = 5, {f_3} = 13, {f_4} = 7, {f_5} = 3

n = \sum f = {f_1} + {f_2} + {f_3} + {f_4} + {f_5} = 2 + 5 + 13 + 7 + 3 = 30

Using the formula of geometric mean for grouped data, geometric mean in this case will become:

G.M{\text{ of }}X = \overline X = \sqrt[n]{{{x_1}^{{f_1}} \cdot {x_2}^{{f_2}} \cdot {x_3}^{{f_3}} \cdot {x_4}^{{f_4}} \cdot {x_5}^{{f_5}}}}

\overline X = \sqrt[{30}]{{{{(13)}^2} \cdot {{(14)}^5} \cdot {{(15)}^{13}} \cdot {{(16)}^7} \cdot {{(17)}^3}}}

\overline X = \sqrt[{30}]{{2.33292 \times {{10}^{35}}}} = {(2.33292 \times {10^{35}})^{\frac{1}{{30}}}}

\overline X = 15.0984 \approx 15.10

The method explained above for the calculation of geometric mean is useful when the numbers of values in given data are small in number and the facility of electronic calculator is available. When a set of data contains large number of values then we need an alternative way for computing geometric mean. The modified or alternative way of computing geometric mean is given as under:

For Ungrouped Data
For Grouped Data
G.M{\text{ of }}X = \overline X = Anti\log \left( {\frac{{\sum \log x}}{n}} \right)
G.M{\text{ of }}X = \overline X = Anti\log \left( {\frac{{\sum f\log x}}{{\sum f}}} \right)

Example:
Find the Geometric Mean of the values 10, 5, 15, 8, 12

x
\log x
10
1.0000
5
0.6990
15
1.1761
8
0.9031
12
1.0792
Total
\sum \log x = 4.8573

G.M{\text{ of }}X = \overline X = Anti\log \left( {\frac{{\sum \log x}}{n}} \right)
\overline X = Anti\log \left( {\frac{{4.8573}}{5}} \right)
\overline X = Anti\log \left( {0.9715} \right)
\overline X = 9.36

Example:

Find the Geometric Mean for the following distribution of students’ marks:

Marks
0 - 30
30 - 50
50 - 80
80 - 100
No. of Students
20
30
40
10

Solution:

Marks
No. of Students
f
Mid Points
x
f\log x
0 - 30
20
15
20\log 15 = 23.5218
30 - 50
30
40
30\log 40 = 48.0168
50 - 80
40
65
40\log 65 = 72.5165
80 - 100
10
90
10\log 90 = 19.5424
Total
\sum f = 100
 
\sum f\log x = 163.6425

G.M{\text{ of }}X = \overline X = Anti\log \left( {\frac{{\sum f\log x}}{{\sum f}}} \right)
\overline X = Anti\log \left( {\frac{{163.6425}}{{100}}} \right)
\overline X = Anti\log \left( {1.6364} \right)
\overline X = 43.29