5.14 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS
Measures of Central Tendency and Measures of Dispersion
- A.M. of the following frequency distribution is 5.6. Find the missing frequency.
x f
2 4
4 2
6 ––
8 3
10 2
[Ans. 4 ] - A.M. of the distribution is
56.46. Find missing frequencies. Daily Wages (
) Frequency
45 5
50 48
55 f 3
60 30
65 f 5
70 8
75 6
Total 150
[Ans. 41, 12]
5.1.2. GEOMETRIC MEAN (G. M.)
Definition. : The geometric mean (G) of the n positive values x 1 , x 2 , x 3 .............xn is the nth root of the
product of the values i.e. G=nx. x ....., x 1 2 n It means, G = (x 1. x2,......... xn )1/ n
Now taking logarithms on both sides, we find
1 2 n 1 n
logG^1 log (x. x .........,x )^1 (logx ..... logx )^1 log x.....(1)
=n =n + + =n∑
∴G = antilog
(^1) log x
n
∑
Thus, from formula (1) we find that the logarithm of the G. M. of x 1 , x 2 ....., xn = A.M. of logarithms of x 1 , x 2 ,
......, x n.
Properties :
- The product of n values of a variate is equal to the nth power of their G. M. i.e., x 1 , x 2 , ......, xn = Gn
(it is clear from the definition)] - The logarithm of G. M. of n observations is equal to the A.M. of logarithms of n observations. [Formula
(1) states it]