The most widely used and popular measure of representing the whole data by one value is what most laymen call an 'average' and what the statisticians call the Arithmetic Mean, its value is obtained by adding together all the items and by dividing this total by the number of items. The Arithmetic mean may be :
(i) Simple arithmetic mean, or
(ii) the Weighted arithmetic mean.
The Calculation of simple arithmetic mean-individual observations
The process of computing mean in case of individual observations (i.e. where frequencies are not provided) is very simple. Add together the various values of the variable and then divide the total by the number of items, symbolically:
X = X1 + X2 + X3 + .... + Xn
N or X = ∑X/N
Illustration:- The following table shows the monthly income of 10 employees in an office.
Income ($) 4,780 5,760 6,690 7,750 4,840 4,920 6,100 7,810 7,050 6,950
Calculate the arithmetic mean of incomes
Let income be denoted by the symbol X.
Solution:
Calculation of arithmetic mean
employee |
Monthly income ($) X |
employee |
Monthly income ($) X |
1 |
4780 |
6 |
4920 |
2 |
5760 |
7 |
6100 |
3 |
6690 |
8 |
7810 |
4 |
7750 |
9 |
7050 |
5 |
4840 |
10 |
6950 |
|
|
N=10 |
∑X = 62,650 |
X = ∑X
N: ∑X = 62,650, N = 10
X = 62,650/10 = 6,265.
Therefore the average income is $, 6,265.
Short-cut method: The arithmetic mean can be calculated by using what is known as the arbitrary method. When deviations are taken from an arbitrary origin, the formula of arithmetic mean is:
X‾= A + Σd/N
Where, A is the assumed mean and d is the deviation of items from the assumed mean, i.e. d = (X - A)
Steps:
1. Take an assumed mean
2. Take the deviations of items from the assumed mean and then denote these deviations by d
3. Now obtain the sum of deviations, i.e. Σd
4. Apply the formula: X‾= A + Σd/N
Solution: X‾= A + Σd/N
A = 7000, Σd = -7350, N = 10
X‾= 7000 - 7350/10 = 7000 - 735 = 6265.
Therefore the average income is $6265
This is true for an ungrouped data. But for the grouped data considerable saving of time is possible by adopting the short-cut method.