Introduction
Weighted mean and weighted standard deviation are used when different observations have different importance (weights or frequencies). Instead of treating every value equally, each observation is multiplied by its weight - common in marks with different weightage, inventory with varying counts, or grouped frequency tables.
This pattern is important because it gives a more accurate central value and spread when data points contribute unequally.
Pattern: Weighted Mean and SD
Pattern
The key concept: Use weights to compute a weighted average; compute weighted squared deviations to find weighted variance and then take square root for weighted SD.
Formulas:
Weighted mean (μ) = (Σ wᵢ × xᵢ) ÷ (Σ wᵢ)
Weighted variance (σ²) = (Σ wᵢ × (xᵢ - μ)²) ÷ (Σ wᵢ)
Weighted standard deviation (σ) = √[ (Σ wᵢ × (xᵢ - μ)²) ÷ (Σ wᵢ) ]
Notes:
• If weights are frequencies, these formulas are identical to grouped-data formulas.
• Use population form (divide by Σw) for aptitude problems unless asked to use sample correction.
Step-by-Step Example
Question
A test has three sections with scores and weights as follows: Section A score = 80 (weight 2), Section B score = 70 (weight 3), Section C score = 90 (weight 1). Find the weighted mean and weighted standard deviation.
Solution
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Step 1: Identify values and weights
Observations x: 80, 70, 90.
Corresponding weights w: 2, 3, 1. -
Step 2: Compute weighted mean μ
Σ wᵢ = 2 + 3 + 1 = 6.
Σ wᵢ × xᵢ = (2×80) + (3×70) + (1×90) = 160 + 210 + 90 = 460.
Weighted mean μ = 460 ÷ 6 = 76.67 (rounded to 2 decimals). -
Step 3: Compute deviations (xᵢ - μ) and squared deviations
For 80: deviation = 80 - 76.67 = 3.33 → squared = 11.09 → weighted = 2 × 11.09 = 22.18.
For 70: deviation = 70 - 76.67 = -6.67 → squared = 44.49 → weighted = 3 × 44.49 = 133.47.
For 90: deviation = 90 - 76.67 = 13.33 → squared = 177.69 → weighted = 1 × 177.69 = 177.69. -
Step 4: Sum weighted squared deviations and compute variance
Σ wᵢ × (xᵢ - μ)² = 22.18 + 133.47 + 177.69 = 333.34.
Weighted variance σ² = 333.34 ÷ Σw = 333.34 ÷ 6 = 55.56. -
Step 5: Weighted standard deviation
Weighted SD σ = √55.56 ≈ 7.45.
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Final Answer:
Weighted mean ≈ 76.67, Weighted SD ≈ 7.45.
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Quick Check:
The weighted mean is pulled more toward Section B (higher weight 3) where score is 70, so mean < 80 and >70 - 76.67 is sensible. SD shows moderate spread given one high (90) and one low (70) value. ✅
Quick Variations
1. If all weights are equal, weighted mean reduces to simple arithmetic mean and weighted SD reduces to usual SD.
2. For grouped frequency tables, treat frequency as weight for each class midpoint.
3. If weights sum to 1 (probability weights), same formulas apply; Σw = 1 simplifies variance denominator.
Trick to Always Use
- Step 1: Always compute Σw first - it is the divisor for both mean and variance.
- Step 2: Compute Σ(w×x) to get weighted mean quickly.
- Step 3: Use Σ(w×(x - μ)²) for variance; don’t forget to multiply squared deviations by their weights before summing.
Summary
Summary
In the Weighted Mean and SD pattern:
- Weighted mean = Σ(w×x) ÷ Σw - weights shift the centre toward more important observations.
- Weighted variance = Σ[w×(x - μ)²] ÷ Σw; weighted SD = sqrt of this variance.
- Frequencies are a special case of weights; use the same steps for grouped data.
- Always multiply squared deviations by weights before summing - this is the most common source of error.
Hint: Compute Σ(w×x) then divide by Σw.
Common Mistakes: Forgetting to sum weights or using equal weights by mistake.