GEOMETRIC MEAN

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.

x₁, x₂, x₃, ............x_n

All the above values are being positive, then the GM of x is given by

G = (x₁ ⋅ x₂ ⋅ x₃ ............ x_n)^1/n

For a grouped frequency distribution, the GM is given by

G = (x₁^f1 ⋅ x₂^f2 ⋅ x₃^f3 ............ x_n^fn)^1/N

where N = ∑f.

Properties of Geometric Mean

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. 

Practice Problems

Problem 1 : 

Find the geometric mean of 2, 4 and 8.

Solution :

Formula to find geometric mean : 

G = (x₁ ⋅ x₂ ⋅ x₃ ............ x_n)^1/n

Fitting the given data in to the above formula, we get

G = (2 ⋅ 4 ⋅ 8)^1/3

= (2⁶)^1/3

= 2²

= 4

Problem 2 : 

Find the geometric mean of 3, 6 and 12.

Solution :

Formula to find geometric mean : 

G = (x₁ ⋅ x₂ ⋅ x₃ ............ x_n)^1/n

Fitting the given data in to the above formula, we get

G = (3 ⋅ 6 ⋅ 12)^1/3

= (6³)^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 = (x₁^f1 ⋅ x₂^f2 ⋅ x₃^f3 ............ x_n^fn)^1/N

= (2² ⋅ 4³ ⋅ 8³ ⋅ 16²)^1/10

= (2² ⋅ 2⁶ ⋅ 2⁹ ⋅ 2⁸)^1/10

= (2^{2 + 6 + 9 + 8})^1/10

= (2²⁵)^1/10

= 2^25/10

= 2^5/2

= (2⁵)^1/2

= √(2⁵)

= √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

= 4√2

Problem 4 :

If A be the arithmetic mean  of two positive unequal quantities X and Y and G be their Geometric mean, then

a)  A < G      b)  A > G     c)  A ≤ G    d)  A ≥ G

Solution : 

For any set of positive observations, we have the following inequality

A.M ≥ G.M ≥ H.M

The equality sign occurs, when all the observations are equal. If all observations are positive and unequal then the inequality is 

A.M > G.M > H.M

Therefore, we can conclude that AM > GM for positive unequal quantities.

Problem 5 :

Geometric mean of three observations 40, 50 and x is 10. The value of x is 

a)  2   b)  4    c)  1/2    d) None

Solution : 

Given geometric mean of three observations = 10

Number of observations = 3

x₁ = 40, x₂ = 50 and x₃ = x

geometric mean = (40 ⋅ 50 ⋅ x)^1/3

10 = (40 ⋅ 50 ⋅ x)^1/3

10³ = 40 ⋅ 50 ⋅ x

1000 = 40 ⋅ 50 ⋅ x

x = 1000/2000

x = 1/2

So, the value of x is 1/2

Problem 6 :

Geometric mean of 8, 4, 2 is

a)  4      b)  2     c)  8    d) none

Solution : 

x₁ = 8, x₂ = 4 and x₃ = 2

geometric mean = (8 ⋅ 4 ⋅ 2)^1/3

= (64)^1/3

= (4³)^1/3

= 4^3x1/3

= 4

So, the required geometric mean is 4, option a is correct.

Problem 7 :

If the AM and HM of two numbers are 6 and 9 respectively, then geometric mean is 

a)  7.35     b)  8.5      c)  6.75      d) none

Solution :

Arithmetic mean = 6

Harmonic mean = 9

Geometric mean = √A.M x H.M

= √6 x 9

= √54

= 7.35

So, the required geometric mean is 7.35.

Problem 8 :

If the Arithmetic mean and geometric mean of 10 observations are both 15, the n the value of Harmonic mean is

a)  less than 15      b)  more than 15

c)  15    d) cannot be determined

Solution :

If both arithmetic mean and geometric mean are 15, it means all the observations are constant. Therefore, harmonic mean also be 15.

Problem 9 :

If two variables a and b are related by c = ab then geometric mean of c is equal to 

a) geometric mean of a + geometric mean b

b) geometric mean of a x geometric mean b

c) geometric mean of a - geometric mean b

d) geometric mean of a / geometric mean b

Solution :

If two variables a and b are related by c = ab, then

= geometric mean of a x geometric mean b

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