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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

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:

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

Where:

• x_i represents each data point
• w_i represents the corresponding weight of each data point
• $\Sigma$ 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

$\text{ Simple Mean }=\frac{\sum x_i}{n}$

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. Multiply each score by its corresponding weight:

85 × 0.30 = 25.5

78 × 0.20 = 15.6

92 × 0.50 = 46

2. Add the weighted scores together:

25.5 + 15.6 + 46 = 87.1

3. Add weight.

0.30 + 0.20 + 0.50 = 1

Using formula

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

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

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. 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

2. Add the weighted grades together:

22 + 23 + 15.2 + 25.5 = 85.7

3. Add weight.

0.20 + 0.25 + 0.20 + 0.30 = 1

Using formula

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

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

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. Multiply each return by its corresponding weight:

8% × 0.50 = 4%

5% × 0.30 = 1.5%

5% × 0.20 = 2.4%

2. Add the weighted returns together:

4 + 1.5 + 2.4 = 7.9%

3. Add weight.

0.50 + 0.30 + 0.20 = 1

Using formula

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

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

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. Multiply the grade points by their corresponding credit hours:

3.7 × 4 = 14.8

3.3 × 3 = 9.9

3.0 × 2 = 6.0

2. Sum the weighted grade points:

14.8 + 9.9 + 6.0 = 30.7

3. Sum the total credit hours:

4 + 3 + 2 = 9

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

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

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. 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

2. Sum the weighted grade points:

19 + 14.4 + 9.6 + 5.6 = 48.6

3. Sum the total credit hours:

5 + 4 + 3 + 2 = 14

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

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

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.

$\text{ Weighted Mean }=\frac{\sum \left(x_i\times w_i\right)}{\sum w_i}$

1. 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

2. Add the weighted grade points:

11.7 + 13.6 + 9.3 + 5.8 + 4.0 = 44.4

3. Sum the total credit hours:

3 + 4 + 3 + 2 + 1 = 13

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

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

Thus, the student’s weighted GPA is 3.42.

5.0Practice Questions on Weighted Arithmetic Mean

1. 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.
2. 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.
3. 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

4. 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.

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.

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:

$\text{ Weighted Mean }=\frac{\sum \left(w_i\times x_i\right)}{\sum w_i}$

Where:

• w_i is the weight of each data point.
• x_i is the value of each data point.