Geometric Mean

The geometric mean is another measure of central tendency based on mathematical footing, like arithmetic mean. The geometric mean can be defined as:

“The geometric mean is the nth positive root of the product of 'n' positive given values.

Hence, the 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:

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} which are repeated {f_1},{f_2},{f_3},...,{f_k} times respectively, then the geometric mean 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 \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 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, the 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 to calculate the geometric mean is useful when the values in the given data are small in number and an electronic calculator is available. When a set of data contains a large number of values then we need an alternate way to compute the geometric mean. The modified or alternative way of computing the geometric mean is given as:

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