Weighted average

A weighted average is an arithmetic mean where each value is multiplied by a weight before summing, and the sum is divided by the total weight rather than the count. Formula:

weighted_mean = Σ(wᵢ × xᵢ) / Σ(wᵢ)

If all weights are equal, the weighted average reduces to the regular mean. If one value has weight zero, it doesn’t affect the result.

Where weighted averages appear:

• GPA (US grade-point average). Course grades weighted by credit hours. A 4-credit A counts more than a 1-credit A.
• Blended interest rates. Borrower with $200k at 6% and $50k at 4% has a weighted rate of (200×6 + 50×4) ÷ 250 = 5.6%.
• Polling and surveys. Demographic groups are weighted to match the population, so an undersampled group counts more per respondent.
• Index funds. Market-cap-weighted indices (S&P 500) weight constituents by their market cap. Equal-weighted variants assign equal weight regardless.
• Moving averages in time series. Exponentially-weighted moving averages weight recent data more heavily than old data.

Common mistakes: using counts where weights are required (averaging percentages without weighting by sample size produces “Simpson’s paradox” results), or weighting by the wrong dimension entirely. When the question is “what’s the average X across the population,” the right weights are the population frequencies, not raw counts of observations.

Simpson’s paradox — the canonical failure mode: the 1973 Berkeley graduate-admissions case is the textbook example. The university appeared to admit men at a much higher rate than women (44% vs 35%) when aggregating across departments — apparent discrimination. But within each department, women were admitted at slightly higher rates than men. The reversal happened because women disproportionately applied to highly-competitive departments with low overall acceptance rates. The unweighted aggregate reversed the within-group pattern. The fix is to compute weighted averages with the right denominator — admissions rate weighted by applicant pool per department — which removes the apparent discrimination. The same trap shows up in clinical-trial subgroup analyses, A/B-test segment reports, and university ranking aggregations. Reference: NIST/SEMATECH e-Handbook — Weighted means.

Worked example: GPA with credit hours

A semester with four courses: A (4.0) in a 4-credit calculus class, B (3.0) in a 4-credit physics class, A− (3.7) in a 3-credit literature class, and A (4.0) in a 1-credit lab. Unweighted mean of grades: (4.0 + 3.0 + 3.7 + 4.0) / 4 = 3.675. Weighted by credits: (4.0×4 + 3.0×4 + 3.7×3 + 4.0×1) / (4+4+3+1) = (16 + 12 + 11.1 + 4) / 12 = 43.1 / 12 ≈ 3.59. The lab’s perfect grade barely moves the average; the 4-credit B drags it down meaningfully. Reporting 3.67 to a graduate committee when your transcript computes 3.59 is the difference between “qualified” and “flagged for review.”

When to weight by something non-obvious

For portfolio returns, the correct weight is dollar exposure (or duration for fixed income), not the count of holdings. For website session metrics, weight by sessions or users rather than treating each page-load record as equal — heavy users disproportionately influence per-event aggregates. For survey results that combine multiple panels, weight by the inverse of each panel’s known bias, not its sample size. Pollster FiveThirtyEight’s methodology documents this in detail — their poll aggregates apply both house-effect weights and recency weights, and the resulting forecasts beat unweighted poll averages by several percentage points in cross-validation.