What is a Weighted Arithmetic Mean?

The Weighted Arithmetic Mean is an average where each data point contributes a different weight or level of importance to the overall mean. It is calculated by multiplying each value by its respective weight, summing these products, and dividing by the total sum of the weights.

How is a Weighted Arithmetic Mean different from a Simple Arithmetic Mean?

The Simple Arithmetic Mean treats all data points equally, while the Weighted Arithmetic Mean assigns different levels of importance to each data point. The simple mean is the sum of all values divided by the number of values, whereas the weighted mean takes into account the weight associated with each value.

When should you use a Weighted Arithmetic Mean?

A Weighted Arithmetic Mean is used when different data points contribute unequally to the overall outcome. For example: Grading systems where different assessments (quizzes, exams, projects) have different weights. Portfolio returns in finance where different assets have different investment proportions. Price indices where products have different shares in the total expenditure.

Can the weights in a Weighted Arithmetic Mean be negative?

Typically, weights are non-negative because they represent proportions or importance. However, in some advanced contexts like financial analysis (e.g., net losses), weights can be negative. In basic applications, weights are usually positive.

What happens if all weights are the same in a Weighted Arithmetic Mean?

If all weights are equal, the Weighted Arithmetic Mean becomes the same as the Simple Arithmetic Mean. This is because each data point contributes equally to the average, making the weighted mean identical to the simple average.

Weighted Arithmetic Mean: Formula, Calculation & Example Problems

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Weighted Arithmetic Mean

In statistics, the arithmetic mean is one of the most fundamental concepts used to determine the average of a set of numbers. However, in many real-world situations, not all values in a dataset are equally important. To address this, we use the weighted arithmetic mean, which assigns different weights to values based on their relative significance.

1.0What is Weighted Arithmetic Mean?

Simple Arithmetic Mean gives equal importance (weight) to each item of the series. But there can be some cases where all the items of a series are not of equal importance. So, for such cases, we calculated Weighted Arithmetic Mean.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

The Weighted Arithmetic Mean is a type of average where each data point is assigned a weight based on its importance or frequency. The formula for this average gives more significance to some numbers over others by multiplying each value by its corresponding weight before calculating the mean.

2.0Formula for Weighted Arithmetic Mean

The formula to calculate the weighted arithmetic mean is:

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Where:

x_i represents each data point
w_i represents the corresponding weight of each data point
$Σ$ denotes the summation over all data points

In simpler terms, multiply each value by its weight, sum them up, and then divide by the total sum of weights.

3.0What is the Difference Between Simple and Weighted Arithmetic Mean?

The simple arithmetic mean assumes that each data point contributes equally to the final average. It is calculated by summing all data points and dividing by the total number of values.

On the other hand, the weighted arithmetic mean takes into account the varying importance or frequency of data points by assigning different weights. This results in a more accurate representation of datasets where some values are more significant than others.

Simple Arithmetic Mean Formula

$Simple Mean = \frac{\sum x _{i}}{n}$

Where:

x_i represents each data point
n is the total number of data points

Key Differences:

In a simple mean, all data points are treated equally.
In a weighted mean, different data points are given different levels of importance based on their weights.

4.0Solved Example on Weighted Arithmetic Mean

Example 1: Suppose a student’s final grade is determined by assignments (worth 30%), mid-term exams (worth 20%), and final exams (worth 50%). The student scored:

Assignments: 85
Mid-terms: 78
Final exam: 92

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply each score by its corresponding weight:

85 × 0.30 = 25.5

78 × 0.20 = 15.6

92 × 0.50 = 46

Add the weighted scores together:

25.5 + 15.6 + 46 = 87.1

Add weight.

0.30 + 0.20 + 0.50 = 1

Using formula

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

$Weighted Mean = \frac{87.1}{1}$

Weighted Mean = 87.1

Thus, the weighted mean score for the student is 87.1.

Example 2: In a class, the final grade is based on homework (25%), projects (25%), mid-term exams (20%), and final exams (30%). A student has the following grades:

Homework: 88
Projects: 92
Mid-terms: 76
Final exam: 85

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply each grade by its corresponding weight:

88 × 0.25 = 22

92 × 0.25 = 23

76 × 0.20 = 15.2

85 × 0.30 = 25.5

Add the weighted grades together:

22 + 23 + 15.2 + 25.5 = 85.7

Add weight.

0.20 + 0.25 + 0.20 + 0.30 = 1

Using formula

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

$Weighted Mean = \frac{85.7}{1}$

Weighted Mean = 85.7

Thus, the student’s final weighted grade is 85.7.

Example 3: A portfolio consists of three stocks with the following weights and returns:

Stock A (weight 50%): 8% return
Stock B (weight 30%): 5% return
Stock C (weight 20%): 12% return

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply each return by its corresponding weight:

8% × 0.50 = 4%

5% × 0.30 = 1.5%

5% × 0.20 = 2.4%

Add the weighted returns together:

4 + 1.5 + 2.4 = 7.9%

Add weight.

0.50 + 0.30 + 0.20 = 1

Using formula

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

$Weighted Mean = \frac{7.9%}{1}$

Weighted Mean = 7.9%

So, the weighted average return of the portfolio is 7.9%.

Example 4: A student’s GPA is calculated based on the following courses with their respective credit hours and grades:

Mathematics: Credits = 4, Grade = 3.7 (A–)
Science: Credits = 3, Grade = 3.3 (B+)
History: Credits = 2, Grade = 3.0 (B)

Calculate the weighted GPA.

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply the grade points by their corresponding credit hours:

3.7 × 4 = 14.8

3.3 × 3 = 9.9

3.0 × 2 = 6.0

Sum the weighted grade points:

14.8 + 9.9 + 6.0 = 30.7

Sum the total credit hours:

4 + 3 + 2 = 9

Divide the total weighted grade points by the total credit hours:

$Weighted GPA = \frac{30.7}{9} = 3.41$

Thus, the student’s weighted GPA is 3.41.

Example 5: A student’s GPA is based on four courses with their respective credit hours and grades:

Physics: Credits = 5, Grade = 3.8 (A)
Chemistry: Credits = 4, Grade = 3.6 (A–)
English: Credits = 3, Grade = 3.2 (B+)
History: Credits = 2, Grade = 2.8 (B–)

Calculate the weighted GPA.

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply the grade points by their respective credit hours:

3.8 × 5 = 19

3.6 × 4 = 14.4

3.2 × 3 = 9.6

2.8 × 2 = 5.6

Sum the weighted grade points:

19 + 14.4 + 9.6 + 5.6 = 48.6

Sum the total credit hours:

5 + 4 + 3 + 2 = 14

Divide the total weighted grade points by the total credit hours:

$Weighted GPA = \frac{48.6}{14} \approx 3.47$

Thus, the student’s weighted GPA is 3.47.

Example 6: A student has completed five courses with the following credit hours and grades:

Biology: Credits = 3, Grade = 3.9 (A)
Mathematics: Credits = 4, Grade = 3.4 (B+)
Economics: Credits = 3, Grade = 3.1 (B)
Sociology: Credits = 2, Grade = 2.9 (B–)
Art: Credits = 1, Grade = 4.0 (A+)

Calculate the student’s weighted GPA.

Solution:

We can calculate the weighted mean using the formula.

$Weighted Mean = \frac{\sum ( x _{i} \times w _{i} )}{\sum w _{i}}$

Multiply each grade by its corresponding credit hours:

3.9 × 3 = 11.7

3.4 × 4 = 13.6

3.1 × 3 = 9.3

2.9 × 2 = 5.8

4.0 × 1 = 4.0

Add the weighted grade points:

11.7 + 13.6 + 9.3 + 5.8 + 4.0 = 44.4

Sum the total credit hours:

3 + 4 + 3 + 2 + 1 = 13

Divide the total weighted grade points by the total credit hours:

$Weighted GPA = \frac{44.4}{13} \approx 3.42$

Thus, the student’s weighted GPA is 3.42.

5.0Practice Questions on Weighted Arithmetic Mean

A class has two assessments: Assignment (40%) and Exam (60%). A student scores 75 in the assignment and 85 in the exam. Calculate the final score.
A portfolio consists of three investments: Investment A (50%), Investment B (30%), and Investment C (20%). The returns are 10%, 6%, and 12%, respectively. Find the portfolio’s weighted average return.
A company wants to calculate the average customer satisfaction score from three surveys conducted in different regions, each weighted according to the number of customers:

Region A: Weight = 50%, Score = 7.5
Region B: Weight = 30%, Score = 8.2
Region C: Weight = 20%, Score = 6.8

Find the weighted mean satisfaction score

A student’s overall score in a subject is calculated based on the following exam weights and scores:

Weekly Quizzes: Weight = 25%, Score = 80
Midterm Exam: Weight = 35%, Score = 75
Final Exam: Weight = 40%, Score = 85

Find the student’s final weighted score.

A student’s GPA is based on four courses with their respective credit hours and grades:

Physics: Credits = 5, Grade = 3.8 (A)
Chemistry: Credits = 4, Grade = 3.6 (A-)
English: Credits = 3, Grade = 3.2 (B+)
History: Credits = 2, Grade = 2.8 (B-)

Calculate the weighted GPA.

6.0Sample Questions on Weighted Arithmetic Mean

1. How do you calculate the Weighted Arithmetic Mean?

Ans: To calculate the Weighted Arithmetic Mean:

Multiply each value by its corresponding weight.
Sum up these products.
Divide the sum by the total of the weights.

Formula:

$Weighted Mean = \frac{\sum ( w _{i} \times x _{i} )}{\sum w _{i}}$

Where:

w_iis the weight of each data point.
x_i is the value of each data point.

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Weighted Arithmetic Mean

1.0What is Weighted Arithmetic Mean?

2.0Formula for Weighted Arithmetic Mean

3.0What is the Difference Between Simple and Weighted Arithmetic Mean?

Simple Arithmetic Mean Formula

4.0Solved Example on Weighted Arithmetic Mean

5.0Practice Questions on Weighted Arithmetic Mean

6.0Sample Questions on Weighted Arithmetic Mean

Table of Contents

Frequently Asked Questions

What is a Weighted Arithmetic Mean?

How is a Weighted Arithmetic Mean different from a Simple Arithmetic Mean?

When should you use a Weighted Arithmetic Mean?

Can the weights in a Weighted Arithmetic Mean be negative?

What happens if all weights are the same in a Weighted Arithmetic Mean?

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