weighted average

weighted average

The weighted average is helpful to make a decision when there are many factors to consider and evaluate. 

Each of the factors is assigned some weights based on their level of importance, and then the weighted average is calculated using a mathematical formula. 

The weights do not have any physical units and are only numbers expressed in percentages, decimals, or integers.

Weighted Average illustration

In this mini-lesson, we shall explore the topic of weighted average, by finding answers to questions like what is the meaning of weighted average, what are the examples of weighted average, and how to find the weighted average.

Lesson Plan


What Is the Meaning of Weighted Average?

The weighted average assigns certain weights to each of the individual quantities. 

The weighted average formula is the summation of the product of weights and quantities, divided by the summation of weights. 

\(\text{Weighted Average} = \dfrac{\sum (\text {Weights} \times \text {Quantities})}{\sum \text {Weights}} \) 

Assigning weights would mean different levels of importance to different quantities.


What Are the Examples of Weighted Average?

A few of the daily life examples would help us to better understand this concept of weighted average.

  • A teacher evaluates a student based on the test marks, project work, attendance, and class behavior. Further the teacher assigns weights to each of the criterion, to makes a final assessment of the performance of the student.

Example - 1 for weighted average

  • A customer's decision to buy or not to buy a product depends on the quality of the product, knowledge of the product, cost of the product, and service by the franchise. Further, the customer assigns weight to each of these criteria and calculates the weighted average.  This will help him in making the best decision.

Example - 2 for Weighted Average

  • For appointing a person for a job, the interviewer looks at his personality, working capabilities, educational qualification, and team working skills. Based on the job profile, these criteria are given different levels of importance(weights) and then the final selection is done.

Example - 3 for Weighted Average


How to Calculate Weighted Average? 

The weighted average formula is the summation of the product of the weights and quantities, divided by the summation of the weights.

\[ \bar x = \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n} \]

\[\bar  x = \frac{\sum w_nx_n}{\sum w_n} \]

Example

Let us understand this calculation of weighted average with the help of an example.

The below table presents the weights of different decision features of an automobile. 

With the help of this information, we need to calculate the weighted average.

Quantity Weight
Safety - 8/10  40%
Comfort - 6/10  20%
Fuel mileage - 5/10 30%
Exterior looks - 8/10 10%

Let us now calculate the final rating of the automobile using the concept of weighted average.

\[ \begin{align} \text{Weighted Average} &= 40\%\times \frac{8}{10} + 20\% \times \frac{6}{10} + 30\% \times \frac{5}{10} + 10\% \times \frac{8}{10} \\ &= 0.4\times 0.8 + 0.2 \times 0.6 + 0.3 \times 0.5 + 0.1 \times 0.8 \\ &= 0.32+0.12 + 0.15 + 0.08 \\ &= 0.67 = 6.7/10 \end{align} \] 

The overall rating of the car is 6.7/10

 
important notes to remember
Important Notes
  1. The weights given to the quantities can be decimals, whole numbers, fractions, or percentages. 
  2. If the weights are given in percentage, then the sum of the percentage should be \(100\% \)
  3. Weighted average for quantities \(x_i \) having weights in percentage \(P_i \% \) is:
    \[\text{Weighted Average} = \sum P_i \% \times x_i \] 

Solved Examples

Example 1

 

 

Sid wants to purchase a mobile phone and checks on the internet and finds the rating of 8/10 for features and 7/10 for durability.

Based on his requirement Sid gives a weightage of 60% to features and 40% for durability.  

How can you help Sid calculate the final rating for the mobile phone?

mobile phones

Solution

The final rating of the mobile phone can be calculated using the concept of weighted average.

Rating for features = 8/10  and rating for durability = 7/10

Weight for features = 60% and weight for durability = 40%

\[\begin{align} \text{Weighted Average} &= 60\% \times \frac{8}{10} + 40\% \times \frac{7}{10}   \\  &= 0.6 \times 0.8 + 0.4 \times 0.7 \\  &= 0.48 + 0.28 \\ &= 0.76 \\  &= \frac{7.6}{10}  \end{align} \]

\(\therefore \) The final rating of the mobile is \(\dfrac{7.6}{10} \) 
Example 2

 

 

The teacher in a class uses the following metric to promote the students to the next grade.  

Criterion for Measurement Weightage
Class Attendance  20%
Project 30%
Assignment Marks 50%

Johnson has attended 80 classes out of 100 classes, got a score of 3.2 out of 5 in the project and 72 marks out of 100 in the assignment. 

The minimum required marks to be promoted to the next grade is 40% marks. 

Can you help the teacher to calculate the final score of Johnson and finalize his promotion?

Solution

Here, we need to take the weighted average of his attendance, project grade, and the marks in the assignment.  

Also, the weightage for each of the criterion is given in the table.

Applying the weighted average formula we have the following solution.

\[\begin{align} \text{Final Score} &= \text{Attendance Weight x Attendance score + Project Weight x} \\&\text{Project grade score + Assignment Weight x Assignment grade score} \\&= 20 \% \times \dfrac{80}{100} + 30 \% \times \dfrac{3.2}{5} + 50 \% \times \dfrac{72}{100} \\&= 0.2 \times 0.8 + 0.3 \times 0.64 + 0.5 \times 0.72 \\&= 0.16 + 0.192 + 0.36 \\&= 0.712 \end{align} \] 

Hence Johnson scores a total of 71.2% marks. 

\(\therefore \) Johnson would be promoted to the next class
Example 3

 

 

Ron has a supermarket and he earns a profit of  $5000 from his groceries, $2000 from vegetables and $1000 from dairy products.  

He wants to predict his profit for the next month.  

He assigns weights of 6 to groceries, 5 to vegetables, and 8 to dairy products.  

Can you help Ron on how to calculate weighted average of his profits?

Supermarket

Solution

Let us first present the profits and the weightage in a table.

Profits Weights
Groceries - $5000 6
Vegetables - $2000 5
Diary Products - $1000 8

Further applying the formula of weighted average to the above data, we have:

\[\begin{align} \text{Weighted Average} &= \frac{6 \times 5000 + 4 \times 2000  + 8 \times 2000}{6 + 4 + 8} \\  &= \frac{30000 + 8000 + 16000}{18} \\ &=\frac{54000}{18}  \\ &= 3000\end{align} \] 

\(\therefore \) The weighted average of the profits is $3000

 

 
Thinking out of the box
Think Tank

What would be the passing marks for a student in a final examination? 

  • The passing marks is to be calculated based on the test score, the project work, and the attendance of the student.
  • The passing marks for the test is 4 out of 10, for the project, it is 2 out of 5 and the student has to attend a minimum of 60 classes out of 100 classes.
  • Also, the teacher has given a weightage of 40% for the test marks, 40% for the class attendance, and 20% for the project.

Interactive Questions 

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of weighted average. The math journey around weighted average starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions

1. How to calculate weighted average?

We use the following weighted average formula for calculation.

\[ \overline x = \frac{w_1x_1 + w_2x_2 + ......+ w_nx_n}{w_1 + w_2 + ... + w_n} \] 

Here \(w_1, w_2, w_3, ...... w_n\) are the weights and \(x_1, x_2, x_3, ....... x_n\) are the quantities.

2. What is weighted average?

The weighted average is the method of calculating the average, in which each of the quantities is assigned a weight.  '

Different weights are assigned to each of the quantities, based no their level of importance.

Weighted average is the summation of the product of the weights and quantities, divided by the summation of the weights.

\[\text{Weighted average} = \frac{\sum w_i.x_i}{\sum w_i} \]

3. What is weighted average calculator?

The weighted average calculator is an automated calculator to help calculate the weighted average.

The input for the weighted average calculator is the quantities, and the weights assigned to these quantities.

And the output answer is the weighted average of those quantities.

4. What is weighted average cost of capital?

The weighted average cost of capital helps to find the capital value of the company.

The capital includes fixed assets, cash in hand, goods, brand value.  All of these are assigned certain weights and the weighted average formula is used to calculate the weighted average cost of capital.

More Important Topics
Numbers
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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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