WEIGHTED AVERAGES EXPLAINED WITH EXAMPLES
Understand weighted averages using practical, everyday examples
What Is a Weighted Average?
A weighted average is a type of mean that takes into account the varying degrees of importance of the numbers in a data set. Unlike a simple arithmetic average — where each value contributes equally — a weighted average multiplies each number by a predetermined weight before summing and dividing by the total weight.
Weighted averages are used extensively in finance, economics, academic grading systems, and data analysis. They help in scenarios where not all values contribute equally to the final calculated average.
Formula for Weighted Average
The general formula for calculating a weighted average is:
Weighted Average = (Σwixi) / Σwi
Where:
- wi = the weight of the i-th item
- xi = the value of the i-th item
- Σ = the summation symbol
This method ensures that items with a higher assigned weight have a greater impact on the final average.
Why Use Weighted Averages?
Weighted averages are especially useful when certain data points are deemed more significant than others. For instance, in a stock portfolio, the performance of stocks you’ve invested more money in should have a greater effect on your portfolio’s return. Similarly, in student assessments, a final exam might count more toward the final grade than a quiz or homework assignment.
In the following sections, we’ll explore practical examples to further illustrate the utility of weighted averages across different fields.
Weighted Averages in Education and Grading
Educational institutions commonly use weighted averages to compute students’ final grades. Different assignments, quizzes, and exams typically carry varying levels of importance, denoted as weights. Here’s how it works.
Example: Calculating a Course Grade
Suppose a student is enrolled in a course where the grading breakdown is as follows:
- Homework: 20%
- Midterm Exam: 30%
- Final Exam: 50%
Let’s assume the student scores:
- Homework: 85%
- Midterm Exam: 70%
- Final Exam: 90%
To calculate the final grade using a weighted average:
Weighted Average = (85 × 0.20) + (70 × 0.30) + (90 × 0.50)
= 17 + 21 + 45
= 83%
Therefore, the student’s final grade is 83%, not the simple average of the three scores (which would be 81.7%). The heavier weight of the final exam significantly impacts the final result.
Why It Matters
Weighted grading reflects the importance the instructor places on different components of a course. It allows assessment to better align with learning outcomes. For example, if a final project is critical to demonstrating overall understanding, it can justifiably carry more weight.
Students also benefit by understanding how their performance in various components affects their overall grade, guiding them to allocate their time and effort wisely.
Multiple Component Evaluation
Beyond academia, this way of evaluating performance is applicable in certifications or courses run by professional bodies. Weighted schemes ensure that stronger emphasis is placed on the more valuable aspects of a curriculum.
In some systems, even different subjects may contribute unequally to a cumulative GPA, depending on credit hours or core requirements. In such cases, weighted averages ensure that grades in more essential or credit-heavy courses dominate the GPA calculation.
Weighted Averages in Finance and Investing
Weighted averages are deeply embedded in the world of finance and investing. They play a critical role in calculating returns, performance metrics, and valuations. Let’s look into several real-world financial applications.
1. Weighted Average Portfolio Return
A common use of weighted averages in investing is to calculate the overall return of a diversified portfolio where each asset has a different value or allocation percentage.
Suppose an investor's portfolio consists of the following holdings:
- Stock A: £10,000, return = 8%
- Stock B: £5,000, return = 12%
- Stock C: £15,000, return = 6%
Total investment = £30,000
Weighted Portfolio Return = [(10,000 × 0.08) + (5,000 × 0.12) + (15,000 × 0.06)] / 30,000
= (800 + 600 + 900) / 30,000
= 2,300 / 30,000
= 7.67%
In this case, the investor’s overall return was 7.67%, not the simple average of the three returns (8.67%). This happens because Stock C had the largest share of the investment and the lowest return, pulling the weighted average down.
2. Weighted Average Cost of Capital (WACC)
WACC is a financial metric used to estimate a firm’s cost of financing, factoring in both debt and equity. Each component is assigned a weight based on its proportion in the company’s capital structure.
Formula:
WACC = (E/V × Re) + [(D/V × Rd) × (1 − Tc)]
Where:
- E = market value of equity
- D = market value of debt
- V = E + D
- Re = cost of equity
- Rd = cost of debt
- Tc = corporate tax rate
WACC helps companies assess whether to go ahead with a project or investment based on its anticipated returns versus cost of capital.
3. Weighted Average Interest Rate
Borrowers carrying multiple loans with different interest rates can calculate the weighted average interest rate to get a clear picture of their overall cost of debt servicing.
For example, consider a consumer with:
- Loan A: £12,000 at 5%
- Loan B: £8,000 at 7%
Weighted Interest Rate = [(12,000 × 0.05) + (8,000 × 0.07)] / 20,000
= (600 + 560) / 20,000
= 1,160 / 20,000
= 5.8%
Using the weighted average, this person is effectively paying 5.8% interest across their total outstanding debt, a more accurate representation than taking the mean of 5% and 7%.
