Weighted average (weighted mean)

Definition and formula

In symbolic form the weighted average is computed as (Σ w_i x_i) / (Σ w_i). Put another way, it is the average of values scaled by importance: the numerator is the sum of the weights times their corresponding values and the result is divided by the total of the weights. If the weights sum to 1 the denominator can be omitted and the weighted average becomes a convex combination of the values.

Key properties

• Linearity: the weighted average is linear in the values when weights are fixed.
• Range: with nonnegative weights that sum to a positive number, the weighted average lies between the minimum and maximum of the values.
• Normalization: weights are often scaled so their sum equals 1, making interpretation as proportions simple.
• Edge cases: if the total weight is zero the expression is undefined; negative weights are allowed mathematically but can produce extrapolation and require careful interpretation.

Uses and examples

Weighted averages appear across many fields. In finance they give the return of a portfolio by weighting asset returns by allocation; in education they combine assessment scores when components carry different credit; in survey analysis they adjust for unequal selection probabilities; and in time series a weighted moving average emphasizes recent observations. Example: with values 70, 80, 90 and weights 1, 2, 1 the weighted average is (1*70+2*80+1*90)/(1+2+1)=80.

Variations and related concepts

Closely related measures include the weighted geometric mean and weighted median, which apply weighting in multiplicative or order-based ways. A weighted moving average applies variable weights over time. Practitioners must choose weights objectively (e.g., sample sizes, probabilities) or transparently when based on expert judgment.

Notes on history and interpretation

The idea extends the simple average to situations where data points have unequal relevance. It is a foundational tool in statistics, economics and engineering for aggregating heterogeneous information. Clear reporting of how weights were assigned is essential for reproducible and interpretable results.