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Weighted Averages – Need, Calculation & Examples ⚖️📊

A weighted average is used when different items in a dataset carry different levels of importance (weights). It is one of the most practical and widely used averages in business, finance, accounting, and economics.

Examples of weights:

• Quantity purchased
• Marks assigned to subjects
• Number of units produced
• Index number weights

Whenever data items do not contribute equally, we must use a weighted average, not a simple average.

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What Is a Weighted Average?

A weighted average gives more importance to some values based on their weights.

Formula

Weighted Average = Σ(Wi × Xi) / ΣWi

Where:

• Xi = value of the ith item
• Wi = weight of the ith item
• ΣWi = sum of weights

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Why Do We Need Weighted Averages?

Simple averages assume all values are equally important, which is rarely true in business.

Weighted averages are needed when:

• Different subjects have different credit weights
• Prices differ based on quantity purchased
• Production units vary across products
• Index numbers require base-year weights

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Types of Weighted Averages

1. Weighted Arithmetic Mean

Used in:

• Cost accounting
• Price indices
• Marks and GPA calculations

2. Weighted Geometric Mean

Used for:

• Index numbers (e.g., Consumer Price Index)
• Growth rate analysis

3. Weighted Harmonic Mean

Used for:

• Average speed when distances differ

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Solved Examples

Example 1 — Weighted Average Marks (GPA type)

A student scores:

• 80 marks in a 3-credit subject
• 70 marks in a 2-credit subject
• 90 marks in a 4-credit subject

Compute the weighted average.

Xi = marks
Wi = credits
Σ(WiXi) = 3×80 + 2×70 + 4×90
         = 240 + 140 + 360 = 740
ΣWi = 3 + 2 + 4 = 9
Weighted Average = 740 / 9 ≈ 82.22

The student's weighted score = 82.22.

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Example 2 — Weighted Average Price

A trader buys:

• 50 units at ₹10 each
• 30 units at ₹15 each
• 20 units at ₹20 each

Weighted Price = Σ(WiXi) / ΣWi
               = (50×10 + 30×15 + 20×20) / (50 + 30 + 20)
               = (500 + 450 + 400) / 100
               = 1350 / 100 = 13.50

Weighted Average Price = ₹13.50 per unit.

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Example 3 — Weighted Average Production

Factory output:

• Product A: 100 units at ₹5 profit
• Product B: 200 units at ₹7 profit

Weighted Profit = (100×5 + 200×7) / (100 + 200)
                = (500 + 1400) / 300
                = 1900 / 300 ≈ 6.33

Average profit per unit = ₹6.33.

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When to Use Weighted Average vs Simple Average

Use simple average when:

• All quantities or importance levels are the same.

Use weighted average when:

• Quantities differ
• Items have different significance
• Values must reflect real impact

Example: A simple average would treat 1 kg and 100 kg purchase as equal — which is incorrect.

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Properties of Weighted Average ⭐

• Uses all observations and weights
• More accurate than simple averages
• Essential for index numbers
• Suitable when values differ in importance
• May change drastically if weights change

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Applications in Business & Finance 📌

• GPA/CGPA calculation (credits as weights)
• Cost accounting (units produced as weights)
• Index numbers (Laspeyres, Paasche)
• Portfolio returns (investment proportions)
• Weighted average cost of capital (WACC)

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Summary ✨

• Weighted average = Σ(WiXi) / ΣWi
• Used when items have different weights
• More realistic than simple averages
• Crucial in business, finance, and economics

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Quiz Time 🎯