Meaning
“Weighted Average” is an average resulting from the multiplication of each component of a series of observation by a factor reflecting its importance. It is a mean in which each item being averaged is multiplied by a number (called ‘weight’) based on the item’s relative importance or number of occurrence or frequencies. The result of such multiplication is summed and the total is divided by the sum of the weight. Thus, it is an average where each quantity to be averaged is assigned a given weight. The weights signify the relative importance of each quantity on the average. So, a weighted average reflects the significance or importance of a series of numbers within the group of numbers.
How to Calculate Weighted Average

Multiply each value with their respective weight (weights may be the frequencies of each value, number of times occurring or importance, etc)

Add up the products of each value items

Add up the total weight

Divide the total value (as in ‘2’ above) by the total weight (as in ‘3’ above).
The Advantages of Weighted Average

The benefit of using a weighted average is that it indicates an average number that reflects the relative importance of each item that is being averaged.

Weighted averages smooth out the wide fluctuations of a series and help to understand the representative value of a series.

It also useful in accounting of uneven data

Another benefit of weighted average is that it assumes that equal values are equivalent in proportion.

Weighted means are more realistic than simple average because it assigns different weights to different values as per their relative importance or occurrence.

They are at the same time, very flexible because you have the option, as a statistician, to multiply the numbers with appropriate weights of your choice to represent a more realistic mean value.

Moreover, it brings clarity in grading and indexing (for example, stock indexing like ABC analysis, or High School or University grading, GPA score, etc)
Example # 1
The marks obtained in Statistics by the students of class eleven in a school are given in the following table. From the following table find out the weighted average or weighted mean of the marks obtained in Statistics.
Marks Obtained 
Number of students 
40 
19 
43 
21 
48 
25 
56 
11 
57 
7 
59 
6 
60 
8 
62 
5 
63 
3 
68 
4 
71 
2 
Solution:
Marks Obtained 
Number of students (Frequency) 
Each Value is Multiplied With Their Respective Weight 
Multiplied Value 
40 
19 
40 X 19 
760 
43 
21 
43 X 21 
903 
48 
25 
48 X 25 
1200 
56 
11 
56 X 11 
616 
57 
7 
57 X 7 
399 
59 
6 
59 X 6 
354 
60 
8 
60 X 8 
480 
62 
5 
62 X 5 
310 
63 
3 
63 X 3 
189 
68 
4 
68 X 4 
272 
71 
2 
71 X 2 
142 
Total Frequency or Weight =111 
Total of Products value = 5625 
Therefore, the weighted average of the marks obtained = 5625 / 111 = 50.676
Answer: 50.676
Example # 2
A class of 30 students took Mathematics Test, 12 students had an arithmetic mean or average score of 75 and remaining students had an average score of 55. What is the average score for the whole class?
Solution
Step 1
Total number of students (sum of all frequency or total weights) = 30
Step 2
In the given problem the weights are 12 (for 75 marks) and 18 (30 — 12) for (for 55 marks).
Step 3
The total marks obtained by 30 students in the class = 75 X 12 + 55 X 18 = 900 + 990 = 1890
Step 4
Therefore, the Weighted Average of the marks = Sum of Weighted Terms / Total weights
= (75 X 12 + 55 X 18) / 30 = 1890 / 30 = 63
Answer: The weighted average score in Mathematics for the whole class is 63
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