Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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B Com : Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

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GEOMETRIC MEAN (G. M.) Definition. :
The geometric mean (G) of the n positive values x1 , x2 , x3 ………….xn is the nth root of the product of the values i.e  Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Now taking logarithms on both sides, we find

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Thus, from formula (1) we find that the logarithm of the G. M. of x1 , x2 ….., xn = A.M. of logarithms of x1 , x2 , …..., xn .

Properties :
1. The product of n values of a variate is equal to the nth power of their G. M. i.e., x1 , x2 , ……, xn = Gn (it is clear from the definition)]
2. The logarithm of G. M. of n observations is equal to the A.M. of logarithms of n observations. [Formula (1) states it]

3. The product of the ratios of each of the n observations to G. M. is always unity. Taking G as geometric mean of n observations x1 , x2 , ……., xn the ratios of each observation to the geometric mean are

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

By definition,   Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev Now the product of the ratios.

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

4. If G1 , G2……, are the geometric means of different groups having observations n1 , n2………respectively, then the G. M. (G) of composite group is given by

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev where N = n1 + n2 + …..i.e., log Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Example 15 : Find the G. M. of the number 4, 12, 18, 26.

Solution :  Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

 

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev 

Weighted Geometric Mean : If f1 , f2 , f3……f n are the respective frequencies of n variates x1 , x2 , x3 ,…….xn , then the weighted G. M. will be

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

 

Now taking logarithm.

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Steps to calculate G. M.  
1. Take logarithm of all the values of variate x.
2. Multiply the values obtained by corresponding frequency
3. Find f log x and divide it by f , i.e., calculate  Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev
4. Now antilog of the quotient thus obtained is the required G. M. The idea given above will be clear from the following example.

Example 16 : Find (weighted) G. M. of the table given below : ––

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Solution :

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics B Com Notes | EduRev

Advantages Geometric Mean
(i) It is not influenced by the extreme items to the same extent as mean.
(ii) It is rigidly defined and its value is a precise figure.
(iii) It is based on all observations and capable of further algebraic treatment.
(iv) It is useful in calculating index numbers.

Disadvantages of Geometric Mean :
(i) It is neither easy to calculate nor it is simple to understand.
(ii) If any value of a set of observations is zero, the geometric mean would be zero, and it cannot be determined.
(iii) If any value is negative, G. M. becomes imaginary. [Use. It is used to find average of rates of changes.]

 

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