Geometric Mean
Here we will learn all the Geometric Mean Formula With Example. The Geometric Mean is n^{th} root of the product of n quantities of the series. It is observed by multiplying the values of items together and extracting the root of the product corresponding to the number of items. Thus, the square root of the products of two items and cube root of the products of the three items are the Geometric Mean.
Usually, GM is never larger than AM. If there are negative numbers and zeros in the series, the GM cannot be used. Logarithms can be used to find GM to reduce the large number and to save time.
The geometric mean (GM) of a series of ‘n’ positive numbers is given by:
1. In case of discrete series without frequency,
\[GM=\sqrt[n]{{{x}_{1}}.{{x}_{2}}…..{{x}_{n}}}\]
It is also given by
\[GM=anti\log (\frac{\sum{\log x}}{n})\]
2. In case of discrete series with frequency,
\[GM=\sqrt[n]{{{x}_{1}}^{{{f}_{1}}}.{{x}_{2}}^{{{f}_{2}}}….{{x}_{n}}^{{{f}_{n}}}}\]
Where,
\[n={{f}_{1}}+{{f}_{2}}+….+{{f}_{n}}\]
It is also given by,
\[GM=anti\log \{\frac{\sum{f\log x}}{n}\}\]
3. In case of continuous series,
\[GM=\sqrt[n]{{{m}_{1}}^{{{f}_{1}}}.{{m}_{2}}^{{{f}_{2}}}….{{m}_{n}}^{{{f}_{n}}}}\]
Where,
\[n={{f}_{1}}+{{f}_{2}}+….+{{f}_{n}}\]
And m_{1}, m_{2}, …, m_{n} are the mid points of class intervals.
It is also given by,
\[GM=anti\log \{\frac{\sum{f\log m}}{n}\}\]
Weighted Geometric Mean
Like the weighted arithmetic mean we can also calculate the weighted geometric mean.
\[{{G}_{W}}=anti\log \{\frac{\sum{W\log x}}{W}\}\]
GW = Weighted Geometric Mean
∑ W log x = Sum of the products of the logarithms of the value x and their corresponding weights.
∑ W = Sum of the weights.
Example 01 
Find the Geometric Mean of data 2, 4, 8.
Solution:
Here x_{1} = 2, x_{2} = 4, x_{3} = 8
\[GM=\sqrt[3]{{{x}_{1}}\times {{x}_{2}}\times {{x}_{3}}}\]
\[GM=\sqrt[3]{2\times 4\times 8}=\sqrt[3]{64}=4\]
Example 02 
Find the GM of following data.
Marks(x)  130  135  140  145  150 
No. of Students(f)  3  4  6  6  3 
Solution:
Marks (x) 
No. of Students (f)  log x  f log x 
130  3  2.113 
6.339 
135 
4  2.130  8.520 
140  6  2.146 
12.876 
145 
6  2.161  12.996 
150  3  2.176 
6.528 

∑ f = n = 22 
∑ f log x = 47.23 
\[GM=anti\log \{\frac{\sum{f\log x}}{n}\}\]
\[=anti\log \{\frac{47.23}{22}\}=140.212\]
Example 03 
Find out GM for given data
Yield of wheat in MT  010  1020  2030  3040  4050  5060 
No. of farms frequency(f)  3  16  26  31  16  8 
Solution:
Class Interval 
Midvalue (m)  No. of farms (f)  log m  f log m 
010  5  3  0.699 
2.097 
1020 
15  16  1.176  18.816 
2030  25  26  1.398 
36.348 
3040 
35  31  1.544  47.864 
4050  45  16  1.653 
26.448 
5060 
55  8  1.740  13.920 
∑ f = n = 100 
∑ f log m = 145.493 
\[GM=anti\log \{\frac{\sum{f\log m}}{n}\}\]
\[=anti\log \{\frac{145.493}{100}\}=28.505\]