Weighted Average Calculator

**Weighted average formula:** WA = Σ(value × weight) / Σ(weights) Or: weighted_sum / total_weight. **Worked example: GPA calculation** Course | Grade | Credits - English | 3.7 (A-) | 3 - Math | 4.0 (A) | 4 - History | 3.0 (B) | 3 - Lab | 3.7 (A-) | 1 Weighted sum: 3.7×3 + 4.0×4 + 3.0×3 + 3.7×1 = 11.1 + 16 + 9 + 3.7 = 39.8. Total credits: 11. GPA = 39.8 / 11 = 3.62. **Worked example: course grade** Component | Score | Weight - Homework | 90 | 20% - Midterm | 75 | 30% - Final | 85 | 50% Weighted: 90×0.20 + 75×0.30 + 85×0.50 = 18 + 22.5 + 42.5 = 83. Sum of weights: 1.0 (100%). Final grade: 83. **Worked example: portfolio return** Holdings: - Stock A: $5,000, returned 15%. - Stock B: $10,000, returned 8%. - Stock C: $20,000, returned 5%. Weighted average return: (15% × 5,000 + 8% × 10,000 + 5% × 20,000) / (5,000 + 10,000 + 20,000) = (750 + 800 + 1,000) / 35,000 = 2,550 / 35,000 ≈ 7.29%. Total portfolio return: 7.29% (closer to dominant Stock C's 5% because of larger holding). **Weighted vs simple average:** | Type | Treats values | |---|---| | Simple average | Equally | | Weighted average | According to weights | **When weights are equal:** If all weights are the same (k), formula reduces to: WA = (kV₁ + kV₂ + kV₃) / (k + k + k) = k(V₁+V₂+V₃) / 3k = (V₁+V₂+V₃)/3. Simple average is special case of weighted (all weights equal). **Common weighting schemes:** | Application | Weight | |---|---| | GPA | Credit hours | | Course grade | Assignment percentage | | Portfolio | Dollar amount invested | | Indices | Market capitalization | | Surveys | Demographic factors | | Quality control | Severity of defect | | Real estate | Property size or value | | Demographic stats | Population proportions | **Worked example: weighted average price** Bought 100 shares of stock at $50 each, then 50 more at $80 each. Average cost per share? Weighted: (50×100 + 80×50) / 150 = (5000 + 4000)/150 = 60. Simple average: (50+80)/2 = 65. Weighted ($60) is lower because more shares bought at $50. **Survey weighting:** If poll over-sampled urban residents: - Adjust by weighting urban responses less. - Weighted average reflects general population. **Real estate:** Average price per sq ft of multiple properties: Don't simple average prices per sq ft (e.g., 200 + 300 + 250 = 250 simple). Instead: weighted by sq ft. total_price / total_sq_ft. **Index funds (S&P 500):** Market-cap-weighted: stocks with bigger market cap have more weight in the index. So Apple (~$3T cap) contributes much more to the index than a smaller company. Other indices: equal-weighted, price-weighted (Dow Jones). **Common pitfalls:** - **Forgetting weights**: just averaging values gives wrong answer. - **Negative weights**: usually don't make sense (except advanced applications). - **All zero weights**: undefined (division by zero). - **Treating percentage weights as values**: keep weight units separate. **Weighted vs harmonic mean:** Weighted is most common, but: - **Harmonic mean**: for rates (speeds, etc.). - **Geometric mean**: for ratios (returns, growth rates). Each has its place; choose based on what makes sense for the values. **Trimmed mean:** Sometimes called "weighted" but is a different concept: - Discard top n% and bottom n% of values. - Compute simple average of remainder. Used to reduce effect of outliers. **Statistical weighting:** In statistics: - **Sample size weighting**: larger samples get more weight. - **Inverse variance weighting**: more precise measurements weighted more. - **Importance sampling**: weights reflect importance distribution. Used in: meta-analysis, statistical estimation, machine learning. **GPA system:** US GPA scale: - A: 4.0 - A-: 3.7 - B+: 3.3 - B: 3.0 - B-: 2.7 - C+: 2.3 - C: 2.0 - D: 1.0 - F: 0.0 Weighted by credit hours. Cumulative GPA tracks performance across all courses. **Course grade calculation:** Common weights: - **Homework**: 10-30% - **Quizzes**: 10-20% - **Midterm**: 20-30% - **Final exam**: 25-40% - **Project**: 10-25% Total: 100%. For each component: percent × your score. Sum all = final percent. **Portfolio weighted return:** R_portfolio = Σ(amount_in_each × return_of_each) / total_portfolio_value. Each holding's return scaled by its share of total portfolio. **Software:** - **Excel**: =SUMPRODUCT(values, weights) / SUM(weights). - **Python (numpy)**: np.average(values, weights=weights). - **R**: weighted.mean(values, weights). - **MATLAB**: sum(values .* weights) / sum(weights). **Pitfalls:** - **Confusing values and weights**: keep separate in formula. - **Negative weights**: usually invalid (except in some statistics). - **Zero total weight**: undefined. - **Treating as simple average**: weights are not optional. - **Mixing units**: ensure all values in same units; weights can be in different units. **Common applications:** - **Academic**: GPA, course grades, transcript averages. - **Financial**: portfolio returns, weighted indices. - **Statistical**: poll/survey weighting. - **Manufacturing**: weighted quality scores. - **Real estate**: average price per sqft across portfolio. - **Surveys**: representative averages. - **Sports analytics**: weighted player stats. **Educational notes:** Weighted averages taught: - Middle school: introduction. - High school: course grades, GPA. - Statistics: theoretical applications. - Finance: investment analysis. Important real-world math skill. **Pitfalls (continued):** - **For percentages**: weights and values both can be percentages — different roles. - **For very small weights**: precision issues. - **For weighted standard deviation**: more complex than simple sd. - **For weighted median**: separate calculation.