How to Calculate Weighted Average?

Weighted Average is a method of finding the average of a set of numbers where each number (or data point) is given a weight based on its importance or relevance. Weighted averages are commonly used in various fields such as finance, economics, education, and statistics, where different data points may have different levels of importance.

This article offers a full explanation of weighted average calculation, including relevant concepts and methods, as well as a few examples based on it.

Weighted average is a statistical measure that considers the varying importance of different elements in a data set. In a standard arithmetic mean (or simple average), each data point contributes equally to the final average value. However, in a weighted average, certain data points have a greater impact on the result than others, depending on their assigned weights.

For Example: Suppose we are calculating the average score of a student’s performance in a course. If exams are considered more important than homework assignments, we would assign higher weights to exam scores and lower weights to homework scores when calculating the overall average.

Weighted averages offer flexibility in analyzing data by allowing us to adjust the impact of different values based on specific criteria or preferences.

The formula for calculating the weighted average (x̄) of a dataset with n values is given by:

[Tex]Weighted Average (\bar{x})= \frac{w_1\cdot x_1 + w_2\cdot x_2 + w_3\cdot x_3 + w_4\cdot x_4 + ……. + w_n\cdot x_n }{w_1 + w_2 + w_3 + w_4 + ……. + w_n } [/Tex]

OR

[Tex] Weighted Average (\bar{x})= \frac{\sum_{i = 1}^{n}w_ix_i }{\sum_{i = 1}^{n}w_i} [/Tex]

Where:

• x_i represents the individual values in the dataset.
• w_i denotes the weights assigned to each value.
• [Tex]\sum_{i = 1}^{n}[/Tex]signifies the summation over all values in the dataset.

Calculating a weighted average involves the following steps:

• Assign Weights: Determine the relative importance or significance of each data point and assign corresponding weights.

• Multiply Values by Weights: Multiply each data point by its corresponding weight.

• Summation: Add all the products from the previous step.

• Sum of Weights: Add all the weights together.

• Divide: Divide the summation of the weighted values by the sum of the weights to obtain the weighted average.

Example1. Calculate the weighted average score of a student based on three exams with different weights:

• For Exam 1: Score = 80, Weight = 30%.
• For Exam 2: Score = 90, Weight = 40%.
• For Exam 3 : Score = 85, Weight = 30%.

Solution:

Using the weighted average formula:

• x̄ = (80 ×0.30 + 90 × 0.40 + 85 × 0.30) / (0.30 + 0.40 + 0.30)
• x̄ = (24 + 36 + 25.5) / (0.30 + 0.40 + 0.30)
• x̄ = 85.5 / 1 = 85.5

So, the weighted average score is 85.5.

Example 2: A student’s semester grades and corresponding credit hours are:

• For Course A let us take : Grade = 3.8, Credit Hours = 3
• For Course B let us take : Grade = 2.5, Credit Hours = 4
• For Course C let us take : Grade = 4.0, Credit Hours = 2

Calculate the student’s GPA (Grade Point Average) for the semester.

Solution:

Using the weighted average calculation process:

• Step 1: Assign Weights: The credit hours serve as the weights.
• Step2: Multiply and Sum: (3.8×3) + (2.5×4) + (4.0×2) = 29.4
• Step3: Sum of Weights: 3+4+2 = 9
• Step4: Divide: 29.4/9 = 3.27 (rounded to two decimal places)

Therefore, the student’s GPA is 3.27.

Example 3: An investor holds three stocks in their portfolio with the following values and weights:

• For Stock X let us take : Value = $5,000, Weight = 40%
• For Stock Y let us take : Value = $8,000, Weight = 35%
• For Stock Z let us take : Value = $2,000, Weight = 25%

Calculate the weighted average return of the portfolio.

Solution:

Using the weighted average calculation process:

• Assign Weights: The portfolio values serve as the weights.
• Multiply and Sum: (5000 x 0.40) + (8000 x 0.35) + (2000 x 0.25) = 5300
• Sum of Weights: 0.40 + 0.35 + 0.25 = 1
• Divide: 5300 / 1 = 5300

Therefore, the weighted average return of the portfolio is $ 5300.

Example 4: A company has the following distribution of salaries and number of employees:

• Fet us assume for Salary Level 1: $40,000, Number of Employees = 15
• Fet us assume for Salary Level 2: $55,000, Number of Employees = 10
• Fet us assume for Salary Level 3: $70,000, Number of Employees = 5

Calculate the weighted average salary of the company.

Solution:

Using the weighted average calculation process:

• Assign Weights: The number of employees serve as the weights.
• Multiply and Sum: (40000 x 15) + (55000 x 10) + (70000 x 5) = 1475000
• Sum of Weights: 15 + 10 + 5 = 30
• Divide: 1475000 / 30 = 49166.67

Therefore, the weighted average salary of the company is $49,166.67.

Example 5: A course grade is calculated based on the following components and their respective weights:

• Let us take values for Homework Assignments: Average = 88%, Weight = 20%
• Let us take values for Quizzes: Average = 92%, Weight = 15%
• Let us take values for Midterm Exam: Score = 85%, Weight = 30%
• Let us take values for Final Exam: Score = 90%, Weight = 35%

Calculate the student’s final weighted course grade.

Solution:

Using the weighted average calculation process:

• Assign Weights: The weights are given as percentages.
• Multiply and Sum: (88 x 0.20) + (92 x 0.15) + (85 x 0.30) + (90 x 0.35) = 88.95
• Sum of Weights: 0.20 + 0.15 + 0.30 + 0.35 = 1 (or 100%)
• Divide: 88.95 / 1 = 88.95

Therefore, the student’s final weighted course grade is 88.95%.

Also Read,

• How to calculate the Weighted Mean?
• Weighted Average Calculator
• Methods to find Mean

Q1. A final grade for a course is calculated using the following components and weights:

• Homework: Average score = 85, Weight = 25%
• Projects: Average score = 90, Weight = 35%
• Midterm: Score = 80, Weight = 20%
• Final Exam: Score = 88, Weight = 20%

Calculate the student’s final weighted grade.

Q2. A company’s employee satisfaction survey assigns different weights to various categories:

• Work Environment: Average score = 7.8, Weight = 50%
• Compensation: Average score = 6.5, Weight = 30%
• Work-Life Balance: Average score = 8.0, Weight = 20%

Calculate the weighted average satisfaction score.

Q3. An investor’s portfolio consists of three assets with different returns and weights:

• Asset A: Return = 12%, Weight = 50%
• Asset B: Return = 8%, Weight = 30%
• Asset C: Return = 15%, Weight = 20%

Calculate the weighted average return of the portfolio.

Q4. A student’s final grade in a class is based on several components with different weights:

• For Attendance we have: Average score = 100, Weight = 10%
• For Assignments we have: Average score = 85, Weight = 30%
• Midterm Exam: Average Score = 78, Weight = 20%
• Final Exam: Average Score = 92, Weight = 40%

Calculate the student’s final weighted grade.

Q5. A firm’s product sales are weighted by their revenue contribution:

• Product X: Sales = $10,000, Weight = 20%
• Product Y: Sales = $25,000, Weight = 50%
• Product Z: Sales = $15,000, Weight = 30%

Calculate the weighted average sales value.

What is the difference between a weighted average and a simple average?

A weighted average is, by definition, a average calculated on the basis of different occurrences. On the other hand, with simple averages all values are equal–nobody can intervene to ensure that some get more emphasis than others.

Can weights in a weighted average be negative?

No, in a weighted average, the weights should be non-negative, as they record the relative importance of each data point.

How is a weighted average used in financial analysis?

In financial analysis, it is very common to have weighted averages. They are used for instance in calculating the returns on investments (or bands of returns) made over time; evaluating financing strategies when interest rates change; or in finding how to turn an overall asset value into portfolio assets.

What if the sum of weights in a dataset is not equal to 1?

The sum of weights in a dataset does not have to be equal to 1. If the weights are expressed as a percentage, then they have to add up to 100%.

Can weighted averages be used for qualitative data?

Weighted average are usually used for quantitative data, but can indeed lean over to the qualitative side or talk along it by taking numerical scores of qualitative categories.