Suppose you take exams in your High School Final for five subjects, Math, English, Physics, Chemistry and History. And the common average score is the arithmetic mean which is got by adding all the five scores and dividing by 5. But for admission into Math honors course, the College administration may attach more importance or weights to Math, English and Physics scores when compared to scores gained in Chemistry and History. In such situations weighted average is computed which represents student's necessary skills and knowledge better. Weighted average can also be calculated for other types of mean like Geometric and Harmonic means.

Weighted average is obtained when each quantity included in the average has a weight assigned to it. For example, a student score 50 in subject A and 70 in subject B out of a maximum marks of 100. Now subject A has a credit of 2 and subject B has a credit 3. If we have to take an average of these two marks we will take it as,

Average = $\frac{1}{5}$ $(2\times 50+3\times 70)$ = 62

Here, we can see that marks of A is multiplies by 2, and that of B by 3. To get the average each quantity is multiplied by its weight and then average is taken by dividing the total by the total weight.
If x1, x2, .....xn are n values with corresponding weights w1, w2,.......wn, then the weighted average of the data set is given by the formula:

X = $\frac{w_{1}x_{1}+w_{2}x_{2}+......+w_{n}x_{n}}{w_{1}+w_{2}+.....w_{n}}$

which can also be written using Sigma notation as

$\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}$


Weighted average formula is similar to mean formula for frequency distribution, where frequencies can be thought of as weights. The steps used for computing the weighted average are similar to those used in computation of arithmetic mean for frequency distributions.
Step 1: Make a table with three columns for Value x, Weight w and Product wx.

Step 2: Enter the values and weights given and complete the third column by finding the product wx for each value.

Step 3: The total under the column weights gives the value for w and the total under the column product gives the value for wx.
 
Step 4: Compute the weighted average using the formula x = $\frac{\sum wx}{\sum w}$
Problem 1: Given below is a table Final scores and the weights attached to it. Find the weighted average for the given scores.

Subject
Scores
x
Weights
w

Math 92
3
Statistics 85 3
English 84
2
Economics 87
2
History 87
1

Let us add the column for the product.

Subject
Scores
x
Weights
w
wx
Math 92
3 276
Statistics 85 3
255
English 84
2
168
Economics 87
2
174
History 87
1 87
Total
$\sum $w = 11$\sum $wx = 960

Weighted Mean of Scores x = $\frac{\sum wx}{\sum w}$

= $\frac{960}{11}$ = 87.3

Problem 2: The average monthly salary of 25 female employees and 35 male employees in a departmental store are
correspondingly 1,200 dollars and 1,500 dollars. Find the average salary of all the employees in the store.

For this problem, number of employees in each category as the weights.

Average salary of all the employees in the store = $\frac{1200\times 25+1500\times 35}{25+35}$

= $\frac{30,000+52,500}{60}$

= $\frac{82500}{60}$ =1375

Hence, average salary of all employees is 1,375 dollars.