FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 5.15- 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 x 1 , x 2 , ......., xn the ratios of each observation to the geometric mean are
x 1
G,
x 2
G......xn
GBy definition, G = nx ,x ,.....,x 1 2 n or, Gn = (x 1 , x 2 , ......, x n). Now the product of the ratios.
x 1
G.x 2
G.......xn
G=1 2 n n
nx .x ....x G 1
G.G.....to n times=G =- If G 1 , G 2 ......, are the geometric means of different groups having observations
n 1 , n 2 .........respectively, then the G. M. (G) of composite group is given by
G=NG .G .... 1 n^12 n^2 where N = n 1 + n 2 + .....i.e., log G =^1 n logG n log G 1 1 2 2 .....
N^ + +^
Example 15 : Find the G. M. of the number 4, 12, 18, 26.
Solution : G =^4 4.12.18.26; here n = 4
Taking logarithm of both sides,
Log G =^14 (log 4 + log 12 + log 18 + log 26)
=^14 (0.6021 + 1.0792 + 1.2553 + 1.4150)
=^14 (4.3516) = 1.0879
∴ G = antilog 1.0879 = 12.25.
5.1.2.1. Weighted Geometric Mean:
If f 1 , f 2 , f 3 ......f n are the respective frequencies of n variates x 1 , x 2 , x 3 ,.......x n, then the weighted G. M. will
be
G = (^123 n)
f f f f 1/n
x 1 ×x 2 ×x 3 ×..... x× n where N = f 1 + f 2 + ......+ fn = ∑f
Now taking logarithm.
Log G =N^1 (f 1 log x 1 + f 2 log x 2 +f 3 log x 3 +.....+ f n log x n)
(^1) f log x.
=N∑ G = anti log
(^1) f log x
N
∑
Steps to calculate G. M.
- Take logarithm of all the values of variate x.
- Multiply the values obtained by corresponding frequency.