Geometric Mean


Statistics / Sunday, August 5th, 2018

Geometric Mean

Here we will learn all the Geometric Mean Formula With Example. The Geometric Mean is nth 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, …, mn 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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