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

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.