Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 5.5

(iii) Method of Assumed mean (by using step deviations)
Table : Calculation of Arithmetic Mean

Class Mid-points

d' =x - A

i f fd’

19.5–29.5 24.5 – 3 5 – 15

29.5–39.5 34.5 – 2 11 – 22

39.5 –49.5 44.5 –1 18 –18

49.5–59.5 54.5=A 0 22 0

59.5–69.5 64.5 1 16 16

69.5 – 79.5 74.5 2 8 16

Total –– –– N = 80 fd'= – 23

A.M. = A + fd ́×i = 54.5 –^23 ×10 = 54.5 - 2.88 = 51.6(approx)

N 80

(^).
5.1.1.1Calculation of A. M. from grouped frequency distribution with open ends
If in a grouped frequency distribution, the lower limit of the first class or the upper limit of the last class are
not known, it is difficult to find the A.M. When the closed classes (other than the first and last class) are of
equal widths, we may assume the widths of the open classes equal to the common width of closed class
and hence determine the AM. But we can find Median or Mode without assumption.
Properties of Arithmetic Mean :

1. The sum total of the values fx is equal to the product of the number of values of their A.M.

e.g. Nx = fx.

2. The algebraic sum of the deviations of the values from their AM is zero.

If x 1 , x 2 ......x n are the n values of the variable x and x their AM then x 1 – x, x 2 – x, ..... x n – x are called

the deviation of x 1 , x 2 ...........xn respectively from from x

Algebraic sum of the deviations ( )

n

= (^) j = 1x - xj
= (x 1 – x ) + (x 2 – x) +..... + (xn – x) = (x 1 + x 2 +....+ x n) – nx = nx – nx = 0
Similarly, the result for a weighted AM can be deduced.

3. If group of n 1 values has AM. x 1 and another group of n 2 values has AM x 2 , then A.M. (x) of the
   composite group (i.e. the two groups combined) of n 1 + n 2 values is given by :
   1 1 2 2
   1 2

x =n x + n x

n + n In general, for a group the AM (x) is given by