Geometric mean is one of the measures of central tendency which can be defined as follows.
For a given set of n positive observations, the geometric mean is defined as the n^{th} root of the product of the observations.
Let the variable x assume n values as given below.
x_{1}, x_{2}, x_{3}, ............x_{n}
All the above values are being positive, then the GM of x is given by
G = (x_{1} ⋅ x_{2 }⋅ x_{3 }............ x_{n})^{1/n}
For a grouped frequency distribution, the GM is given by
G = (x_{1}^{f1 }⋅ x_{2}^{f2} ⋅ x_{3}^{f3 }............ x_{n}^{fn})^{1/N}
where N = ∑f.
1. Logarithm of G for a set of observations is the AM of the logarithm of the observations; i.e.
logG = (1/r)∑logx
2. If all the observations assumed by a variable are constants, say K > 0, then the GM of the observations is also K.
3. GM of the product of two variables is the product of their GM‘s i.e. if z = xy, then
GM of z = (GM of x) ⋅ (GM of y)
4. GM of the ratio of two variables is the ratio of the GM’s of the two variables i.e. if z = x / y, then
GM of z = (GM of x)/(GM of y)
5. Like arithmetic mean, GM also possess some mathematical properties.
6. It is rigidly defined.
7. It is based on all the observations.
8. It is difficult to comprehend.
9. It is difficult to compute.
10. It has limited applications for the computation of average rates and ratios and such like things.
Problem 1 :
Find the geometric mean of 2, 4 and 8.
Solution :
Formula to find geometric mean :
G = (x_{1} ⋅ x_{2 }⋅ x_{3 }............ x_{n})^{1/n}
Fitting the given data in to the above formula, we get
G = (2 ⋅ 4 ⋅ 8)^{1/3}
= (2^{6})^{1/3}
= 2^{2}
= 4
Problem 2 :
Find the geometric mean of 3, 6 and 12.
Solution :
Formula to find geometric mean :
G = (x_{1} ⋅ x_{2 }⋅ x_{3 }............ x_{n})^{1/n}
Fitting the given data in to the above formula, we get
G = (3 ⋅ 6 ⋅ 12)^{1/3}
= (6^{3})^{1/3}
= 6
Problem 3 :
Find the geometric mean for the following distribution :
x : 2 4 8 16
f : 2 3 3 2
Solution :
Formula to find geometric mean for a grouped frequency distribution :
G = (x_{1}^{f1 }⋅ x_{2}^{f2} ⋅ x_{3}^{f3 }............ x_{n}^{fn})^{1/N}
= (2^{2} ⋅ 4^{3} ⋅ 8^{3} ⋅ 16^{2})^{1/10}
= (2^{2} ⋅ 2^{6} ⋅ 2^{9} ⋅ 2^{8})^{1/10}
= (2^{2 + }^{6 + }^{9 + }^{8})^{1/10}
= (2^{25})^{1/10}
= 2^{25}^{/10}
= 2^{5/2}
= (2^{5})^{1/2}
= √(2^{5})
= √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)
= 4√2
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