In case of ungrouped data where weights are involved, our approach for calculating arithmetic mean will be different from the one used earlier.
Example 2.1: Suppose a student has secured the following marks in three tests:
Midterm test 30
Laboratory 25
Final 20
The simple arithmetic mean will be 30 + 25 + 20 / 3 = 25
However, this will be wrong if the three tests carry different weights on the basis of their relative importance. Assuming that the weights assigned to the three tests are:
Midterm test 
2 points 
Laboratory 
3 points 
Final 
5 points 
Solution: On the basis of this information, we can now calculate a weighted mean as shown below:
Table 2.1: Calculation of a Weighted Mean
Type of Test 
Relative Weight (w) 
Marks (x) 
(wx) 
Midterm 
2 
30 
60 
Laboratory 
3 
25 
75 
Final 
5 
20 
100 
Total 
å w = 10 

235 
x = å wx / å w = w_{1} x_{1} + w_{2} x_{2} + w_{3} x_{3 / w1 + w2 + w3}
= 60 + 75 + 100 / 2 + 3 + 5 = 23.5 marks
It will be seen that weighted mean gives a more realistic picture than the simple or unweighted mean.
Example 2.2: An investor is fond of investing in equity shares. During a period of falling prices in the stock exchange, a stock is sold at Rs 120 per share on one day, Rs 105 on the next and Rs 90 on the third day. The investor has purchased 50 shares on the first day, 80 shares on the second day and 100 shares on the third' day. What average price per share did the investor pay?
Solution:
Table 2.2: Calculation of Weighted Average Price
Day 
Price per Share (Rs) (x) 
No of Shares Purchased (w) 
Amount Paid (wx) 
1 
120 
50 
6000 
2 
105 
80 
8400 
3 
90 
100 
9000 
Total 
 
230 
23,400 
Weighted average =
w1 x1 + w2 x2 + w3 x3 / w1 + w2 + w3= å wx/ å w
=6000 + 8400 + 9000 /50 + 80 + 100= 101.7 marks
Therefore, the investor paid an average price of Rs 101.7 per share.
It will be seen that if merely prices of the shares for the three days (regardless of the number of shares purchased) were taken into consideration, then the average price would be
Rs.120 + 105 + 90 / 3= 105
This is an unweighted or simple average and as it ignores thequantum of shares purchased, it fails to give a correct picture. A simple average, it may be noted, is also a weighted average where weight in each case is the same, that is, only 1. When we use the term average alone, we always mean that it is an unweighted or simple average.