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Weighted Average

Introduction

A weighted average, also known as a weighted mean, helps decision-making when various factors are considered and measured. A mathematical formula is used to get the weighted average after each component is given weights based on how significant they are.

Let’s explore the concept of weighted average by learning what it means, looking at some real-world situations, and applying the formula to solve a few examples.

What is the weighted average?

Definition

The term “weighted average” refers to an average in which each quantity is to be averaged given a weight. A weighted average computation accounts for the varying numbers’ significance levels in a data collection. When calculating a weighted average, a specified weight is multiplied by each value in the data set before the final computation is completed.

A weighted average, as opposed to a simple average, considers the relative weight or contribution of the items being averaged. A weighted average computation accounts for the varying numbers’ significance levels in a data collection. When calculating a weighted average, a specified weight is multiplied by each value in the data set before the final computation is completed.

When the numbers in a data collection are given the same weight, a simple average can be less accurate than a weighted average.

The Weighted Average Formula

Compared to the basic average, the weighted average formula is more expressive and descriptive since the resulting average number indicates the relative value of each observation. Unlike the arithmetic mean, the weighted average gives some data points in the data set more weight than others. It can be stated as follows:

$Weighted\: Average=\frac{Sum\: of\: the\: Weighted\: Terms}{Total\: Number\: of\: Terms}$

The weighted average method is utilized in various settings, including financial firms, statistical analysis, and classrooms. Instead of using the simple average alone, a user can better understand a set of facts by using a weighted average. The weight you assign to particular variables in the data set will affect how accurate the results you get using this strategy.

How to Calculate Weighted Average?

When there are numerous factors to consider and assess, the weighted average, also known as the weighted mean, helps to make a decision. The weighted average is determined by applying various weights to each component according to their respective relevance. The weighted average gives each of the different quantities a specific weight. The weights are essentially numerical expressions expressed as percentages, decimals, or integers; they do not correspond to any physical units. 

There are two methods for weighted average calculation. The first method is used when the weights add up to one, whereas the second method is used when they do not. Method 2 calculates the weighted average by multiplying each value by its weight, adding the products, and then dividing the sum of the products by the total weights.

Method 1: When the Weights Add Up to One

Step 1: Arrange the number you want to average.
Step 2: Identify the weight of each data point.
Step 3: Multiply each number by each weighing factor.
Step 4: To get the weighted average, add the data in Step 3.

For instance, you want to calculate your weighted average in Geometry. Your overall course grades are 88, 92, and 95 for the quizzes, performance tasks, and final exams. Assume that the quizzes are worth 25%, the performance assignments are worth 45%, and the final exam is worth 30% of the total weight.

Let us follow the steps to calculate the weighted average.

Step 1: Arrange the number you want to average.

The numbers that must be arranged refer to the grades you obtained in your quizzes, performance tasks, and final exam.

ComponentGrades
Quizzes88
Performance Tasks92
Final Exam95

Step 2: Identify the weight of each data point.

Once you have all the data, you’ll need to know how much each number contributes to your final average. In this example, the quizzes are for 25%, the performance assignments are worth 45%, and the final exam is worth 30%. Thus, the weights add up to 1 or 100% (25% + 45% + 30%). Convert the percentages to decimal form to identify the weighting factors. 

ComponentGradesWeighing Factors
Quizzes880.25
Performance Tasks920.45
Final Exam950.30

Step 3: Multiply each number by each weighing factor.

Once you have arranged your data, multiply each number by the correct weighing factor.

ComponentGradesWeighing Factors
Quizzes880.25
Performance Tasks920.45
Final Exam950.30

Step 4: To get the weighted average, add the data in Step 3.

Here is the math in getting the weighted average,

Weighted Average = ( 88 ) ( 0.25 ) + ( 92 ) ( 0.45 ) + ( 95 ) ( 0.30 ) 
Weighted Average = 22 + 42.40 + 28.50
Weighted Average = 91.90

ComponentGradesWeighing FactorsGrades × Weighing Factor
Quizzes880.2522
Performance Tasks920.4541.40
Final Exam950.3028.50

Weighted Average: 91.90

Hence, your grade in the course Geometry is 91.90. 

Method 2: When the Weights Do Not Add Up to One

Step 1: Arrange the number you want to average.
Step 2: Identify the weight of each data point.
Step 3: Find the sum of all the weights.
Step 4: Multiply the numbers by their respective weights, then add up the results.
Step 5: To find the weighted average, divide the result in Step 4 by the sum of all the weights.

For example, you want to calculate the average time you spend traveling to school for 20 days. The table below shows the numbers you recorded.

Time SpentNumber of Days
15 minutes4
20 minutes8
25 minutes6
30 minutes2

Step 1: Arrange the number you want to average.

Let us refer to the table presented above in our computation. The data collected has been arranged already. The first column indicates the time you spent traveling to school, which are 15 minutes, 20 minutes, 25 minutes, and 30 minutes.

Step 2: Identify the weight of each data point.

According to the data, you traveled to school on four days in which you spent 15 minutes, eight days in which you spent 20 minutes, six days in which you traveled in 25 minutes, and two days in which you traveled for 30 minutes. These number of days would be the weighting factors.

Time SpentNumber of Days(Weighing Factors)
15 minutes4
20 minutes8
25 minutes6
30 minutes2

Step 3: Find the sum of all the weights.

Simply add up all the weights in this example. Hence, we have, 4 + 8 + 6 + 2 = 20 days.

Time SpentNumber of Days(Weighing Factors)
15 minutes4
20 minutes8
25 minutes6
30 minutes2

Sum of all weights = 20

Step 4: Multiply the numbers by their respective weights, then add up the results.

Multiply each number (time spent) by the correct weighing factor (number of days), then add up all the results.

Time SpentNumber of Days
(Weighing Factors)
Time Spent × Weighing Factor
15 minutes460
20 minutes8160
25 minutes6150
30 minutes260
Sum of all weights =20Sum of (time spent x weighing factor) = 430

Step 5: To find the weighted average, divide the result in Step 4 by the sum of all the weights.

Divide the result by the total weights after multiplying each number by its weighting factor and adding the results. Hence, we have,

Weighted Average = 430 / 20 
Weighted Average = 21.50 minutes.

Therefore, your travel time to school averages 21.50 minutes for 20 days. 

Let us use the formula to calculate the result as an alternative solution.

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( (15 × 4) (20 × 8) (25 × 6) (30 × 2) ) / (4 + 8 + 6 + 2)
Weighted Average = (60 + 160 + 150 + 60) / 20
Weighted Average = 430 / 20
Weighted Average = 21.5
Therefore, the weighted average is 21.5.

Why would you use a weighted average?

All numbers are given equal consideration and weight when calculating a simple average, also known as an arithmetic mean. However, a weighted average applies weights that predetermine the proportional significance of each data point.

For example, a teacher evaluates a student based on exam results, project work, attendance, and classroom behavior. The teacher also gives weights to each category to create a final evaluation of the student’s performance. The weights assigned to each criterion help the instructor in her student assessments.

Examples

Example 1

Find the weighted average for the following data sets.

( a ) 5 weighted at 20%, 7 weighted at 65%, and 8 weighted at 15%.
( b ) 10 weighted 25, 12 weighted 30, 16 weighted 20, and 15 weighted at 5.

Solution

( a ) 5 weighted at 20%, 7 weighted at 65%, and 8 weighted at 15%.

Let us arrange the data in a table and use the decimal value as the weighing factor. Hence, we will have 0.20 for 20%, 0.65 for 65%, and 0.15 for 15%.

NumbersWeighing FactorsNumber × Weighing Factor
50.201
70.654.55
80.151.2

Weighted Average = 6.75

Applying the values to the formula we have,

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( (5 × 0.20) (7 × 0.65) (8 × 0.15) ) / (0.20 + 0.65 + 0.15)
Weighted Average = (1 + 4.55 + 1.2) / 1
Weighted Average = 6.75 / 1
Weighted Average = 6.75
Therefore, the weighted average is 6.75.

( b ) 10 weighted 25, 12 weighted 30, 16 weighted 20, and 15 weighted at 5.

Let us arrange the data in a table as shown below. We must multiply each number by its respective weights to start the solution. Then add the results and divide by the sum of all the weights.

NumbersWeighing FactorsNumber × Weighing Factor
1025250
1230360
1620320
15575
Sum of Weighing Factors = 80Sum of (Number x Weighing Factor) = 1005

Applying the values to the formula we have,

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( (10 × 25) (12 × 30) (16 × 20) (15 × 5) ) / (25 + 30 + 20 + 35)
Weighted Average = (250 + 360 + 320 + 75) / 80
Weighted Average = 1005 / 80
Weighted Average = 12.5625
Therefore, the weighted average is 12.5625.

Example 2

A manufacturer buys 3000 units at $2 each, 6,000 at $1.5 each, 5000 at $1.3 each, and 1200 at $1.2 each of a product. Calculate the weighted average.

Solution

Let us use the table below to look at the given data.

Number (Cost per unit)Weighing FactorNumber × Weighing Factor
$230006000
$1.560009000
$1.350006500
$1.212001440

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( ($2 × 3000) ($1.5×3000) ($1.3×5000) ($1.2×1200) ) / (3000 + 6000 + 5000 + 1200)
Weighted Average = (6000 + 9000 + 6500 + 1440) / 15200
Weighted Average = 22940 / 15200
Weighted Average = $1.51

Example 3

Using the grade sheet below, calculate Joseph’s final grade. 

GradesWeighing Factor
Quizzes ( 85)25%
Assignments (90)15%
Laboratory (94)25%
Exams (92)35%

Solution

To calculate Joseph’s final grade, let us multiply each grade by its respective weighing factor, then add the results. By converting the percentages to decimals, we will have 0.25 for 25%, 0.15 for 15%, and 0.35 for 35%.

GradesWeighing FactorGrade × Weighing Factor
Quizzes ( 85)0.2521.25
Assignments (90)0.1513.5
Laboratory (94)0.2523.5
Exams (92)0.3532.2

We may add the results in the third column to calculate the weighted average.

Weighted Average = 21.25 + 13.5 + 23.5 + 32.2 = 90.45.

As for an alternative solution, let us plug the values into the formula.

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( (85 × 0.25) (90 × 0.15) (94 × 0.25) (92 × 0.35) ) / (0.25 + 0.15 + 0.25 + 0.35)
Weighted Average = (21.25 + 13.5 + 23.5 + 32.2) / 1
Weighted Average = 90.45 / 1
Weighted Average = 90.45
Therefore, Joseph’s final grade is 90.45.

Example 4

About the data below, calculate, on average, how much an employer pays an employee an hour of work.

EmployeesRateWorking Hours
A$10 / hour20
B$12 / hour30
C$20 / hour25
D$16 / hour15

Solution

To calculate the weighted average, multiply the rate of each employee by their respective working hours and add the results, then divide the result by the sum of the working hours.

EmployeeRateWorking HoursRate × Working Hours
A$10 / hour20200
B$12 / hour30360
C$20 / hour25500
D$16 / hour15240

Let us plug the values into the formula.

Weighted Average = (Sum of the Weighted Terms) / (Total Number of Terms)
Weighted Average = ( (10 × 20) (12 × 30) (20 × 25) (16 × 15) ) / (20 + 30 + 25 + 15)
Weighted Average = (200 + 360 + 500 + 240) / 90
Weighted Average = 1300 / 90
Weighted Average ≈14.44
Hence, the average is approximately $14.44.

Summary

The term “weighted average” refers to an average in which each quantity is to be averaged given a weight. A weighted average computation accounts for the varying numbers’ significance levels in a data collection. When calculating a weighted average, a specified weight is multiplied by each value in the data set before the final computation is completed.

Compared to the basic average, the weighted average formula is more expressive and descriptive since the resulting average number indicates the relative value of each observation. Unlike the arithmetic mean, the weighted average gives some data points in the data set more weight than others. It can be stated as follows:

$Weighted\: Average=\frac{Sum\: of\: the\: Weighted\: Terms}{Total\: Number\: of\: Terms}$

Steps in Calculating Weighted Average

Method 1: When the Weights Add Up to One

Step 1: Arrange the number you want to average.
Step 2: Identify the weight of each data point.
Step 3: Multiply each number by each weighing factor.
Step 4: To get the weighted average, add the data in Step 3.

Method 2: When the Weights Do Not Add Up to One

Step 1: Arrange the number you want to average.
Step 2: Identify the weight of each data point.
Step 3: Find the sum of all the weights.
Step 4: Multiply the numbers by their respective weights, then add up the results.
Step 5: To find the weighted average, divide the result in Step 4 by the sum of all the weights.

Frequently Asked Questions on Weighted Average (FAQs)

What distinguishes the weighted average from the simple average?

A weighted average, referred to as a weighted mean, requires more calculation than a simple arithmetic mean. A weighted average, as its name suggests, is one in which the many numbers you’re calculating have various weights or values in relation to one another.

A weighted average computation accounts for the varying numbers’ significance levels in a data collection. When calculating a weighted average, a specified weight is multiplied by each value in the data set before the final computation is completed.

When the numbers in a data collection are given the same weight, a simple average can be less accurate than a weighted average.

How to solve the weighted average?

To solve or calculate the weighted average, you may consider the following methods:

Method 1: When the Weights Add Up to One

To find the weighted average, multiply each number by its respective weights and then add the results.

For example, specific weights are given to each component in grade computation to get the final grade.

ComponentsGradesWeighing FactorsGrades × Weighing Factor
Quizzes870.3530.45
Assignments930.2523.25
Final Exam910.4036.4

Weighted Average: 90.10

Weighted Average = ( (87 × 0.35) (93 × 0.25) (91 × 0.40) ) / (0.35 + 0.25 + 0.40 )
Weighted Average = (30.45 + 23.25 + 36.4 ) / 1 
Weighted Average = 90.10 / 1 
Weighted Average = 90.10 

Method 2: When the Weights Do Not Add Up to One

To find the weighted average, multiply each number by its respective weights and get the sum. Then, divide the results by the sum of all the weights. 

For example, Maria wants to calculate her average time spent playing online games for 30 days. The data is presented below.

Time Spent 
(Playing with Online Games)
Number of Days
(Weighing Factors)
Time Spent × Weighing Factor
2 hours48
3 hours515
4 hours728
5 hours630
6 hours848
          Sum of all weights = 30Sum of (time spent x weighing factor) = 129

Weighted Average = ( (2 × 4) (3 × 5) (4 × 7) (5 × 6) (6 × 8) ) / (4 + 5 + 7 + 6 + 8 )
Weighted Average = ( (8) (15) (28) (30) (48) ) / 30
Weighted Average = 129 / 30
Weighted Average = 4.3

Therefore, Maria’s average time on playing online games is 4.3 hours.

What is the formula to calculate the weighted average?

Compared to the basic average, the weighted average formula is more expressive and descriptive since the resulting average number indicates the relative value of each observation. Unlike the arithmetic mean, the weighted average gives some data points in the data set more weight than others. It can be stated as follows:

$Weighted\: Average=\frac{Sum\: of\: the\: Weighted\: Terms}{Total\: Number\: of\: Terms}$

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