Weighted Average
Overview
When observations carry unequal importance, the arithmetic mean misrepresents them, and the weighted form corrects the imbalance: x̄_w = Σ(w_i x_i) ÷ Σw_i. Grade point average is the canonical case. A student earning 4.0 in a one-credit seminar and 3.0 in a four-credit core course has an unweighted mean of 3.5 but a weighted average of (4×1 + 3×4) ÷ 5, or 3.2 — the figure the transcript reports. The same structure computes portfolio returns weighted by position size, inventory cost under the weighted-average-cost convention, and survey results weighted to match census demographics. Weights must be non-negative and their sum non-zero; when they are already normalised to one, the denominator disappears and the expression reduces to a simple dot product.
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $x_i$ | Observation | any | The i-th value being averaged. |
| $w_i$ | Weight | dimensionless | Relative importance of the i-th observation; must be non-negative. |
| $x̄_w$ | Weighted mean | same as observation | Sum of weighted values divided by the sum of weights. |
When Observations Carry Unequal Importance
An arithmetic mean treats every observation as equally informative. Where that assumption fails, the weighted mean restores proportionality:
$$\bar{x}w = \frac{\sum$$} w_i x_i}{\sum_{i} w_i
Grade point average is the textbook instance. A student earning 4.0 in a one-credit seminar and 3.0 in a four-credit core course has an unweighted mean of 3.5 but a weighted average of $(4 \times 1 + 3 \times 4)/5 = 3.2$ — the figure the transcript records.
Normalised Weights
When the weights already sum to one, the denominator vanishes and the expression reduces to $\sum w_i x_i$, a plain dot product. Portfolio returns are usually stated this way because position sizes are naturally expressed as fractions of total value. Converting arbitrary weights to normalised form simply divides each by their sum.
The Weighted Mean as a Centre of Mass
Placing masses $w_i$ at positions $x_i$ along a rod puts the balance point exactly at $\bar{x}_w$. The mechanical analogy explains an otherwise surprising property: the weighted mean always lies between the smallest and largest observation regardless of how extreme the weights become, because a balance point cannot fall outside the masses that define it.
Related Weighted Statistics
The same weights extend to dispersion. Weighted variance replaces the count in the denominator with the sum of weights, and reliability weighting — setting $w_i$ to the inverse of each observation's variance — produces the minimum-variance estimate used in meta-analysis.
Derivation & History
The weighted mean is the value $m$ minimising the weighted sum of squared deviations $\sum w_i (x_i - m)^2$. Differentiating with respect to $m$ and setting the result to zero gives $-2\sum w_i(x_i - m) = 0$, hence $\sum w_i x_i = m \sum w_i$ and the formula follows directly. That least-squares origin is why the weighted mean is the maximum-likelihood estimator of a common mean when each observation is normally distributed with variance proportional to $1/w_i$.
Worked Examples
Semester grade point average
- Weighted sum: 4.5×3 + 3.5×3 + 4.0×2 + 2.5×1 = 13.5 + 10.5 + 8.0 + 2.5 = 34.5
- Sum of weights: 3 + 3 + 2 + 1 = 9 credits
- Divide: 34.5 ÷ 9 = 3.833
Result: GPA 3.83
Portfolio return
- Weights already sum to 1, so the denominator is 1
- 0.60×8 + 0.30×2 + 0.10×(−5) = 4.8 + 0.6 − 0.5 = 4.9
Result: 4.9% portfolio return
Edge Cases & Limitations
Weights summing to zero: The mean is undefined. This arises when weights are allowed to be negative, as with short positions, and requires a different estimator.
Negative weights: The centre-of-mass guarantee fails and the result can fall outside the range of the observations.
Zero weights: Legitimate and simply excludes an observation, but a set of entirely zero weights leaves nothing to average.
Correlated observations: Inverse-variance weighting assumes independence. Correlated inputs require the full covariance matrix rather than scalar weights.
Real-World Applications
Transcripts compute GPA by credit hours. Stock indices weight constituents by market capitalisation, and bond indices by issue size. Consumer price indices weight categories by household expenditure shares. Inventory accounting uses the weighted-average-cost method, and opinion polls weight raw responses to match census demographics.