All About Geometric Mean with Formula and Examples

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 m₁, m₂, …, 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 x₁ = 2, x₂ = 4, x₃ = 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	0-10	10-20	20-30	30-40	40-50	50-60
No. of farms frequency(f)	3	16	26	31	16	8

Solution:

Class Interval	Mid-value (m)	No. of farms (f)	log m	f log m
0-10	5	3	0.699	2.097
10-20	15	16	1.176	18.816
20-30	25	26	1.398	36.348
30-40	35	31	1.544	47.864
40-50	45	16	1.653	26.448
50-60	55	8	1.740	13.920
		∑ f = n = 100		∑ f log m = 145.493

\[GM=anti\log \{\frac{\sum{f\log m}}{n}\}\]

\[=anti\log \{\frac{145.493}{100}\}=28.505\]

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