Weighted Average
A weighted average is a mean that assigns different levels of importance to values in a dataset. Instead of treating every observation equally (as in a simple arithmetic mean), each value is multiplied by a weight that reflects its relative significance. The weighted average better represents situations where some items contribute more to the overall result than others.
Key points
- Calculated as (Σ weight × value) ÷ (Σ weights).
- Useful when data points have different importance, frequency, or relevance.
- Common in finance (cost basis, portfolio returns, WACC, VWAP), inventory valuation, grading systems, and many other fields.
- Introduces subjectivity through the choice of weights and can be sensitive to weight changes.
Formula and calculation
Weighted average = (Σ w_i × x_i) / (Σ w_i)
where x_i are the values and w_i are the corresponding weights. If weights are normalized to sum to 1, the denominator becomes 1 and the formula reduces to Σ w_i × x_i.
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Example with three data points:
Values: 3, 5, 7
Weights: 1, 2, 1
Weighted average = (1×3 + 2×5 + 1×7) / (1+2+1) = (3 + 10 + 7) / 4 = 20 / 4 = 5
Practical examples
-
Stock purchase (cost basis):
An investor buys 100 shares at $10 and later 50 shares at $40.
Total cost = 100×$10 + 50×$40 = $1,000 + $2,000 = $3,000
Total shares = 150
Weighted average price = $3,000 / 150 = $20 per share -
Portfolio return:
A portfolio with 55% stocks (10% return), 40% bonds (5% return), and 5% cash (2% return):
Weighted return = 0.55×10% + 0.40×5% + 0.05×2% = 5.5% + 2.0% + 0.1% = 7.6% -
Inventory valuation:
Weighted average cost method smooths price fluctuations by weighting unit costs by quantities purchased. It contrasts with FIFO and LIFO, which prioritize timing rather than weighted value. -
Corporate finance and trading:
Weighted Average Cost of Capital (WACC) weights the cost of equity and debt by their market-value proportions. Volume-weighted average price (VWAP) weights trade prices by trading volume. Exponential moving averages (EMAs) assign declining weights to older data.
Advantages
- Reflects the relative importance or frequency of observations, producing a more meaningful summary when values differ in significance.
- Reduces the influence of outliers when appropriate weights are applied.
- Flexible—can be tailored to many contexts and objectives.
Disadvantages
- Weight selection can be subjective and introduce bias.
- Results can be sensitive to small changes in weights or inputs, reducing stability.
- Interpretation can be more complex than a simple average; transparency about how weights were chosen is essential.
How it differs from other means
- Arithmetic mean: treats all values equally (simple average). Best when all observations are equally important and distribution is symmetrical.
- Geometric mean: uses the nth root of the product of values and is suited to multiplicative processes (e.g., compound growth rates). It gives equal weight to relative percentage changes rather than absolute values.
- Weighted mean: blends the characteristics above by explicitly incorporating importance through weights.
When to use a weighted average
Use a weighted average when:
* Different observations carry different importance (e.g., varying quantities, population representation, investment weights).
* You need a summary measure that accounts for frequency or significance.
Avoid using weighted averages when weights are arbitrary or cannot be justified, or when transparency about weight choice is required but not available.
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Conclusion
Weighted averages provide a practical and flexible way to summarize data when observations are not equally important. They improve accuracy and relevance in many applications but require careful, transparent selection of weights because that choice directly affects the result.