Weighted average is one of the most commonly employed measures in statistical data to find the average of quantities when each quantity has a certain weight. Let’s have a look at the definition and formula of weighted average here.

What is the weighted average?

Weighted average is an average in which each quantity to be averaged is assigned a weight. These weightings determine the relative importance of each quantity on average. Weightings are the equivalent of having that many like items with the same value involved in the average.

Formula for Weighted average

Let x_i be the observations and w_i be the weights of the observations; the formula of the weighted average is given below.

This can also be written as:

\[\large \overline{x}=\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}\]

Here,

x̄ or W = Weighted average

n = Number of terms to be averaged

w_i = Weights applied to x values

x_i = Data values to be averaged

In simple terms, we can write the above formula as:

\[\large Weighted\;Average=\frac{Sum\;of\;Weighted\;Terms}{Total\;Number\;of\;Terms}\]

To find the weighted term, multiply each term by its weighting factor, which is the number of times each term occurs.

Also, try: Weighted Mean Calculator

Solved Example

Example 1: A class of 25 students took a science test. 10 students had an average score of 80. The other students had an average score of 60. What is the average score of the whole class?

Solution:

Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then add them up.

80 × 10 + 60 × 15 = 800 + 900 = 1700

i.e. Sum of weighted terms = 1700

Step 2: Total number of terms = Total number of students = 25

Step 3: Using the formula, 

= 1700/25

= 68

Answer: The average score of the whole class is 68.

Example 2:

Calculate the weighted average for the following data:

Solution:

From the given,

∑w_ix_i = 4 + 14 + 15 + 18 = 51

∑w_i = 1 + 2 + 3 + 2 = 8

Weighted average = (∑w_ix_i)/∑w_i

= 51/8

= 6.375

Therefore, the weighted average of the given data is 6.375.