A weighted mean is an average that assigns different levels of importance, or weights, to individual values before calculating the result. Weighted mean differs from a standard arithmetic mean, which treats every value equally regardless of its significance to the overall result.
Weighted Mean Formula
Weighted Mean = Σ(w × x) / Σw
- x: Each value
- w: The weight assigned to each value
- Σ(w × x): The sum of each value multiplied by its weight
- Σw: The sum of all weights
Unlike a standard mean, which divides the sum of values by the count of values, weighted mean divides the sum of weighted values by the sum of the weights themselves.
When to Use Weighted Mean
Weighted mean is used when individual values in a data set carry different levels of importance. Weighted mean applies to 3 common scenarios:
- Calculating a course grade where assignments, midterms, and final exams contribute different percentages to the final result
- Calculating portfolio returns where different investments represent different proportions of total capital
- Calculating average product ratings where reviews carry different weights based on reviewer credibility or purchase verification
Standard arithmetic mean is used instead of weighted mean when all values in a data set carry equal importance, such as a simple list of test scores from a single exam.
How to Calculate Weighted Mean
Calculating weighted mean requires 4 steps:
- Assign a weight to each value, typically expressed as a percentage or proportion
- Multiply each value by its corresponding weight
- Sum all weighted values
- Divide the sum of weighted values by the sum of the weights
Weighted Mean: Worked Example (Course Grade)
Scenario: A student’s final grade is calculated from 3 components: homework worth 20%, a midterm exam worth 30%, and a final exam worth 50%.
| Component | Score (x) | Weight (w) | Weighted Value (w × x) |
| Homework | 85 | 0.20 | 17.00 |
| Midterm Exam | 78 | 0.30 | 23.40 |
| Final Exam | 92 | 0.50 | 46.00 |
- Sum of weighted values: 17.00 + 23.40 + 46.00 = 86.40
- Sum of weights: 0.20 + 0.30 + 0.50 = 1.00
- Weighted Mean: 86.40 / 1.00 = 86.40
The student’s final grade is 86.40, a result that differs from the simple arithmetic mean of the 3 scores (85.00), since weighted mean accounts for the unequal contribution of each component.
Weighted Mean vs Arithmetic Mean
| Attribute | Arithmetic Mean | Weighted Mean |
| Formula | Σx / n | Σ(w × x) / Σw |
| Value importance | Equal for all values | Varies based on assigned weight |
| Common use case | Simple averages, such as test scores from one exam | Grades, portfolio returns, weighted survey results |
| Divisor | Count of values (n) | Sum of weights (Σw) |
Comparing the arithmetic mean and weighted mean for the same data set often produces different results, since arithmetic mean ignores the relative importance of each value.
Data set: 85, 78, 92 (unweighted)
Arithmetic Mean: (85 + 78 + 92) / 3 = 255 / 3 = 85.00
Weighted Mean (from the example above): 86.40
Verifying the arithmetic mean portion of this calculation in a statistics calculator confirms the unweighted average of 85.00, which highlights the 1.40-point difference introduced by weighting the final exam more heavily than homework and the midterm.
Weighted Mean for GPA Calculation
Grade Point Average (GPA) is a common application of weighted mean, where each course grade is weighted by the number of credit hours assigned to that course.
Weighted Mean GPA Formula: Σ(grade points × credit hours) / Σ(credit hours)
Example:
| Course | Grade Points (x) | Credit Hours (w) | Weighted Value (w × x) |
| Course A | 4.0 | 3 | 12.0 |
| Course B | 3.0 | 4 | 12.0 |
| Course C | 3.7 | 3 | 11.1 |
- Sum of weighted values: 12.0 + 12.0 + 11.1 = 35.1
- Sum of credit hours: 3 + 4 + 3 = 10
- Weighted GPA: 35.1 / 10 = 3.51
A course with more credit hours contributes more heavily to the final GPA than a course with fewer credit hours, since credit hours function as the weight in this calculation.
Weighted Mean for Investment Portfolios
Weighted mean applies to portfolio return calculations, where each investment’s return is weighted by the proportion of total capital allocated to that investment.
Example:
| Investment | Return (x) | Portfolio Weight (w) | Weighted Value (w × x) |
| Stock A | 8% | 0.50 | 4.00% |
| Stock B | 12% | 0.30 | 3.60% |
| Stock C | −2% | 0.20 | −0.40% |
Weighted Portfolio Return: 4.00% + 3.60% + (−0.40%) = 7.20%
An unweighted arithmetic mean of the same 3 returns produces 6.00%, a result that misrepresents the portfolio’s actual performance, since it ignores the fact that 50% of the portfolio’s capital sits in Stock A.
Common Mistakes When Calculating Weighted Mean
Three mistakes frequently occur when calculating weighted mean:
- Using arithmetic mean instead of weighted mean when values carry unequal importance, producing an inaccurate average
- Failing to ensure weights sum to 1.00 (or 100%), which distorts the final result
- Assigning weights inconsistently, such as mixing percentage weights with raw count weights within the same calculation
Frequently Asked Questions
What is the difference between mean and weighted mean?
Mean treats every value equally when calculating an average, while weighted mean assigns different levels of importance to different values before calculating the result.
Do weights have to add up to 100%?
Yes, when weights are expressed as percentages, they must sum to 100% (or 1.00 as a decimal) for the weighted mean formula to produce an accurate result.
What is a real-world example of a weighted mean?
Course grade calculations, portfolio return calculations, and GPA calculations using credit hours are common real-world examples of weighted mean.
Can weighted mean and arithmetic mean produce the same result?
Yes. Weighted mean equals arithmetic mean when every value receives an identical weight, since equal weighting removes the distinction between the two calculations.
Why does GPA use weighted mean instead of arithmetic mean?
GPA uses weighted mean because courses carry different credit hour values, and a course with more credit hours represents a larger portion of a student’s academic workload.
Is weighted mean more accurate than arithmetic mean?
Weighted mean is more accurate than arithmetic mean, specifically when individual values carry different levels of importance. Arithmetic mean remains accurate and appropriate when all values carry equal importance.