Geometric Mean - Measures of Central Tendency, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

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 

Now taking logarithms on both sides, we find

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

Properties :
1. The product of n values of a variate is equal to the nth power of their G. M. i.e., x₁ , x₂ , ……, x_n = 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 , ……., x_n the ratios of each observation to the geometric mean are

By definition,   Now the product of the ratios.

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

where N = n1 + n2 + …..i.e., log 

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

Solution : 

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

Now taking logarithm.

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 
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 : ––

Solution :

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.]