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:

Mid-term 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:

 Mid-term 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) Mid-term 2 30 60 Laboratory 3 25 75 Final 5 20 100 Total å w = 10 235

x = å wx / å w  w1 x1 + w2 x2 + w3 x3 /    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 the-quantum 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.