STATISTICSArithmeticMathematics Calculator
📊

Weighted Average: Σ(w×x)/Σ(w)

Weighted average = Σ(w_i×x_i)/Σw_i. Higher weight = more influence. GPA: grade×credit hours. Portfolio return: return×weight. Equal weights → arithmetic mean.

Concept Fundamentals
Σ(w×x)/Σw
x̄_w
Grade×credits
GPA
Return×weight
Portfolio
→ arithmetic mean
Equal w

Did our AI summary help? Let us know.

x̄_w = Σ(w×x)/Σw. Equal weights → arithmetic mean. GPA: (grade₁×credits₁ + ...)/(credits₁ + ...). Portfolio: (return₁×weight₁ + ...)/Σweights.

Key quantities
Σ(w×x)/Σw
x̄_w
Key relation
Grade×credits
GPA
Key relation
Return×weight
Portfolio
Key relation
→ arithmetic mean
Equal w
Key relation

Ready to run the numbers?

Why: Not all items are equally important. GPA weights by credit hours. Portfolio return weights by investment. Survey weights by demographic. Weighted average reflects importance.

How: Multiply each value by its weight, sum. Divide by sum of weights. x̄_w = Σ(w_i×x_i)/Σw_i. Weights can be counts, percentages, or importance scores.

x̄_w = Σ(w×x)/Σw. Equal weights → arithmetic mean.GPA: (grade₁×credits₁ + ...)/(credits₁ + ...).
Sources:NCTM StandardsKhan Academy

Run the calculator when you are ready.

Calculate Weighted AverageEnter values and weights

Enter Values & Weights

weighted_avg.sh
CALCULATED
$ weighted_avg --values=90, 85, 70 --weights=0.4, 0.35, 0.25
Weighted Average
83.25
Simple Average
81.6667
Σ(value × weight)
83.25
Σ(weights)
1
Weighted Average Calculator
Weighted Avg = 83.25
Simple Avg: 81.6667 | Difference: 1.5833
numbervibe.com
Share:

Values vs Averages

Contribution by Pair

📐 Step-by-Step Breakdown

SETUP
Values90, 85, 70
Weights0.4, 0.35, 0.25
CALCULATION
Products90×0.4=36, 85×0.35=29.75, 70×0.25=17.5
Sum of products36 + 29.75 + 17.5 = 83.25
Sum of weights0.4 + 0.35 + 0.25 = 1
RESULT
Weighted Average83.25 ÷ 1 = 83.25
COMPARISON
Simple Average81.6667
Difference1.5833

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📊

x̄_w = Σ(w×x)/Σw

— Weighted average

✓

Equal weights → arithmetic mean

— Special case

📋 Key Takeaways

  • • Weighted average = Σ(wᵢ × xᵢ) / Σ(wᵢ). Each value contributes proportionally to its weight.
  • • When all weights are equal, weighted average equals the simple arithmetic mean.
  • • Weighted average is always between the min and max values in the set.
  • • Multiplying all weights by a constant does not change the result.
  • • Weights need not sum to 1; the formula normalizes automatically.

💡 Did You Know?

📚GPA is a weighted average: each course grade weighted by credit hours.Source: Education
📈Stock indices like the S&P 500 use market-cap weighted averages.Source: Finance
💰CPI (Consumer Price Index) weights goods by typical spending shares.Source: Economics
📊Survey results often use weighted averages to match demographic proportions.Source: Statistics
💼Portfolio returns use dollar amounts or allocation percentages as weights.Source: Investing
🥘Recipe scaling uses ingredient ratios as weights for proportional mixing.Source: Cooking

📖 How It Works

Multiply each value by its weight, sum those products, then divide by the sum of weights. Example: values 10, 20, 30 with weights 1, 2, 3 → (10×1 + 20×2 + 30×3) / (1+2+3) = 140/6 ≈ 23.33. The weighted average gives more influence to values with larger weights.

📝 Worked Example: 90, 85, 70 w/ 0.4, 0.35, 0.25

Products: 90×0.4=36, 85×0.35=29.75, 70×0.25=17.5

Sum of products: 36 + 29.75 + 17.5 = 83.25

Sum of weights: 0.4 + 0.35 + 0.25 = 1

Weighted avg: 83.25 / 1 = 83.25

Simple avg = (90+85+70)/3 = 81.67. Final exam (70) has less weight.

🚀 Real-World Applications

📚 Grades & GPA

Course grades weighted by credit hours.

📈 Portfolio Returns

Asset returns weighted by allocation.

📊 Survey Weighting

Responses weighted by sample size.

💰 Price Indices

CPI, PPI use expenditure weights.

🏆 Sports Stats

Performance weighted by games played.

🔬 Research

Meta-analysis uses study weights.

⚠️ Common Mistakes to Avoid

  • Mismatched counts: Values and weights must have the same length.
  • Negative weights: Weights should be non-negative for typical use.
  • Percentage weights: If using %, ensure they sum to 100.
  • Confusing with simple avg: When weights differ, results differ.
  • Wrong weights: Use correct weights for context (credits, dollars, etc.).

🎯 Expert Tips

💡 Percentage Weights

Use weights that sum to 100 for grades and allocation. Calculator normalizes.

💡 Dollar Weights

Use dollar amounts or counts for portfolio and survey weighting.

💡 Compare Both

Compare weighted vs simple average to see how weights shift the result.

💡 Equal Weights

When all weights equal, weighted avg = simple avg.

📊 Reference Table

TermMeaning
xᵢValue i
wᵢWeight for value i
Σ(wᵢ × xᵢ)Sum of (value × weight)
Σ(wᵢ)Sum of weights

📐 Quick Reference

83.25
90,85,70 @ 40,35,25%
20
10,20,30 @ 1,1,1
23.33
10,20,30 @ 1,2,3
Σwx/Σw
Formula

🎓 Practice Problems

90, 85, 70 w/ 0.4, 0.35, 0.25 → 83.25
10, 20, 30 w/ 1, 1, 1 → 20 (equals simple avg)
5, 7, 3 w/ 10k, 5k, 15k → Portfolio return
85, 78, 92 w/ 20%, 30%, 50% → Course grade

❓ FAQ

Do weights need to sum to 1 or 100%?

No. The formula divides by the sum of weights, so any non-negative weights work. Percentages that sum to 100 are common for grades.

When does weighted average equal simple average?

When all weights are equal. Each value contributes equally.

Can weights be negative?

Mathematically possible but rarely used. Negative weights can produce values outside the data range.

How is GPA a weighted average?

Each course grade is multiplied by its credit hours; the sum is divided by total credit hours.

What is the difference from arithmetic mean?

Arithmetic mean treats all values equally. Weighted average gives more influence to values with larger weights.

When to use weighted vs simple average?

Use weighted when some values matter more (e.g., final exam vs homework, large investments vs small).

What if values and weights have different counts?

They must match. Each value needs a corresponding weight.

📌 Summary

Weighted average = Σ(wᵢ×xᵢ)/Σ(wᵢ). Use when values have different importance. GPA, portfolio returns, and survey results commonly use weighted averages. When weights are equal, it equals the simple mean.

✅ Verification Tip

Weighted average always lies between min and max of values. If weights sum to 1, weighted avg = sum of (value × weight). Compare with simple average to sanity-check.

🔗 Next Steps

Explore the Arithmetic Mean Calculator, Geometric Mean for growth rates, or Average Calculator for mean, median, mode.

⚠️ Disclaimer: This calculator provides educational results. For academic grading, financial analysis, or research, verify methodology with your institution or professional standards.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Arithmetic Mean Calculator

Calculate the arithmetic mean (average) of a set of numbers. Simple mode for basic mean; advanced mode adds weighted mean and trimmed mean. Step-by-step...

Mathematics

Average Calculator

Calculate mean, median, mode, and range of a set of numbers. Advanced: weighted average, geometric mean. Step-by-step solutions, bar and doughnut charts....

Mathematics

Geometric Mean Calculator

Calculate geometric mean GM = (x1×x2×...×xn)^(1/n) of positive numbers. Compare with arithmetic and harmonic mean. Essential for investment returns, CAGR...

Mathematics

Harmonic Mean Calculator

Calculate the harmonic mean of a set of positive numbers. HM = n/(1/x₁+...+1/xₙ). Compare with arithmetic and geometric means. Ideal for average speed (same...

Mathematics

Root Mean Square Calculator

Calculate RMS = sqrt(mean of x²). AC voltage equivalent, compare with arithmetic mean. Step-by-step for statistics and signal processing.

Mathematics

Standard Deviation Calculator

Calculate standard deviation (σ or s) from a data set. Choose population or sample. Get mean, variance, and coefficient of variation. Free online calculator...

Mathematics