Weighted Average Formula
Learn the weighted average formula for calculating means where some values count more than others, with practical examples.
The Formula
The weighted average is a calculation where each value in a data set is multiplied by a predetermined weight that reflects its relative importance. Unlike a simple arithmetic mean where every value contributes equally, a weighted average gives some values more influence over the final result than others.
This concept appears constantly in everyday life, even when people do not realize they are using it. When a university course calculates your final grade, the exam might count for 60% while homework counts for 40%. That is a weighted average. When financial analysts calculate a stock index like the S&P 500, each company's stock price is weighted by its market capitalization. When scientists combine measurements taken with instruments of different precision, they weight each measurement by the inverse of its variance to get the most accurate estimate.
The formula works by multiplying each data value by its corresponding weight, summing all those products, and then dividing by the total of all weights. If all weights are equal, the weighted average reduces to the ordinary arithmetic mean. The weights can represent any measure of importance: percentages, frequencies, sample sizes, reliability scores, or arbitrary priority levels. The only requirement is that the weights must be positive numbers.
Weighted averages are also critical in economics for calculating price indices like the Consumer Price Index (CPI), in portfolio management for determining expected returns, and in machine learning for combining predictions from multiple models. The concept extends naturally to continuous distributions, where it becomes an integral rather than a sum, connecting directly to the expected value in probability theory.
Variables
| Symbol | Meaning |
|---|---|
| x₁, x₂, ..., xₙ | The data values being averaged |
| w₁, w₂, ..., wₙ | The weights assigned to each value (must be positive) |
| n | The number of data values |
| Σ wᵢxᵢ | The sum of each value multiplied by its weight |
| Σ wᵢ | The sum of all weights |
Example 1: Course Grade Calculation
Problem: A student scores 92 on the final exam (50% weight), 85 on the midterm (30% weight), and 78 on homework (20% weight). What is the course grade?
Multiply each score by its weight: (92 × 0.50) + (85 × 0.30) + (78 × 0.20)
= 46.0 + 25.5 + 15.6 = 87.1
Sum of weights: 0.50 + 0.30 + 0.20 = 1.00
Weighted average = 87.1 / 1.00 = 87.1 (note: the simple average would be 85.0)
Example 2: Investment Portfolio Return
Problem: A portfolio has $10,000 in stocks (8% return), $5,000 in bonds (4% return), and $2,000 in savings (2% return). What is the weighted average return?
Weights are the dollar amounts: w₁ = 10000, w₂ = 5000, w₃ = 2000
Σ wᵢxᵢ = (10000 × 8) + (5000 × 4) + (2000 × 2) = 80000 + 20000 + 4000 = 104000
Σ wᵢ = 10000 + 5000 + 2000 = 17000
Weighted average return = 104000 / 17000 = 6.12%
When to Use It
Use the weighted average whenever the values in your data set have different levels of importance or frequency.
- Academic grading — combining scores from assignments, exams, and projects with different weights
- Financial analysis — calculating portfolio returns, price indices, or cost of capital
- Survey analysis — adjusting responses based on demographic representation
- Scientific measurement — combining results from instruments with different precision levels
- Machine learning — ensemble methods that combine multiple model predictions