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Weighted Mean and SD

Introduction

Weighted mean और weighted standard deviation तब use किए जाते हैं जब अलग-अलग observations की importance (weights या frequencies) अलग-अलग हो। हर value को बराबर treat करने के बजाय, हर observation को उसके weight से multiply किया जाता है - यह तरीका marks में अलग weightage, inventory में अलग counts, या grouped frequency tables में बहुत common है।

यह pattern इसलिए important है क्योंकि unequal contribution वाले data points के लिए यह ज़्यादा accurate central value और spread देता है।

Pattern: Weighted Mean and SD

Pattern

मुख्य concept: Weights के आधार पर average निकालें; फिर weighted squared deviations के आधार पर weighted variance और SD निकालें।

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:
• अगर weights frequencies हों, तो यही formulas grouped-data formulas बन जाते हैं।
• Aptitude questions में population form (divide by Σw) use किया जाता है, जब तक खास तौर पर sample correction न पूछा जाए।

Step-by-Step Example

Question

एक test में तीन sections के scores और उनके weight इस प्रकार हैं: Section A score = 80 (weight 2), Section B score = 70 (weight 3), Section C score = 90 (weight 1)। Weighted mean और weighted SD निकालें।

Solution

  1. Step 1: Values और weights पहचानें

    Observations x: 80, 70, 90
    Weights w: 2, 3, 1

  2. Step 2: Weighted mean μ निकालें

    Σwᵢ = 2 + 3 + 1 = 6
    Σ(wᵢ × xᵢ) = (2×80) + (3×70) + (1×90) = 160 + 210 + 90 = 460
    Weighted mean μ = 460 ÷ 6 = 76.67 (2 decimal तक)

  3. Step 3: Deviations (xᵢ - μ) और उनके squares निकालें

    80: deviation = 3.33 → squared = 11.09 → weighted = 2 × 11.09 = 22.18
    70: deviation = -6.67 → squared = 44.49 → weighted = 3 × 44.49 = 133.47
    90: deviation = 13.33 → squared = 177.69 → weighted = 1 × 177.69 = 177.69

  4. Step 4: Weighted squared deviations का sum और variance निकालें

    Σ(wᵢ × (xᵢ - μ)²) = 22.18 + 133.47 + 177.69 = 333.34
    Weighted variance σ² = 333.34 ÷ 6 = 55.56

  5. Step 5: Weighted standard deviation निकालें

    σ = √55.56 ≈ 7.45

  6. Final Answer:

    Weighted mean ≈ 76.67, Weighted SD ≈ 7.45

  7. Quick Check:

    Weight 3 Section B पर है (score 70), इसलिए weighted mean नीचे की ओर shift होता है - 76.67 बिल्कुल expected range में है। SD moderate है क्योंकि एक value low (70) और एक high (90) है। ✅

Quick Variations

1. अगर सभी weights equal हों, तो weighted mean simple arithmetic mean बन जाता है और weighted SD usual SD।

2. Grouped frequency tables में weights frequencies होते हैं - midpoint को x मानकर यही formulas apply होते हैं।

3. अगर weights probabilities हों (Σw = 1), तब भी यही formulas use होते हैं - बस denominator पहले से 1 होता है।

Trick to Always Use

  • Step 1: सबसे पहले Σw निकालें - mean और variance दोनों इसी पर depend करते हैं।
  • Step 2: Σ(w×x) जल्दी calculate करें और weighted mean निकालें।
  • Step 3: Variance के लिए squared deviations को उनके weight से multiply करना बिल्कुल न भूलें - यही सबसे common mistake है।

Summary

Summary

In the Weighted Mean and SD pattern:

  • Weighted mean = Σ(w×x) ÷ Σw - weights ज़्यादा important observations की ओर centre को खींचते हैं।
  • Weighted variance = Σ[w×(x - μ)²] ÷ Σw; weighted SD = variance का square root।
  • Frequencies weights की एक special case हैं - grouped data में यही formulas लगते हैं।
  • Squared deviations को weight से multiply करना सबसे crucial step है।

Practice

(1/5)
1. Find the weighted mean of scores 80, 70, 90 with weights 2, 3 and 1 respectively.
easy
A. 76.67
B. 77.50
C. 75.00
D. 74.33

Solution

  1. Step 1: Identify values and weights

    Scores: 80, 70, 90. Weights: 2, 3, 1.

  2. Step 2: Compute Σ(w×x) and Σw

    Σ(w×x) = (2×80) + (3×70) + (1×90) = 160 + 210 + 90 = 460.
    Σw = 2 + 3 + 1 = 6.

  3. Step 3: Weighted mean

    Weighted mean = 460 ÷ 6 = 76.666… ≈ 76.67.

  4. Final Answer:

    76.67 → Option A.

  5. Quick Check:

    Result lies between lowest (70) and highest (90) and is pulled toward the score with larger weight (70) - sensible ✅

Hint: Compute Σ(w×x) then divide by Σw.
Common Mistakes: Forgetting to sum weights or using equal weights by mistake.
2. Given grouped data midpoints 10, 20, 30 with frequencies 2, 3, 5 respectively, find the mean (use frequencies as weights).
easy
A. 22.00
B. 23.00
C. 21.50
D. 24.00

Solution

  1. Step 1: Treat frequencies as weights

    Values x: 10, 20, 30. Weights f: 2, 3, 5.

  2. Step 2: Compute Σ(f×x) and Σf

    Σ(f×x) = (2×10) + (3×20) + (5×30) = 20 + 60 + 150 = 230.
    Σf = 2 + 3 + 5 = 10.

  3. Step 3: Weighted mean

    Mean = 230 ÷ 10 = 23.00.

  4. Final Answer:

    23.00 → Option B.

  5. Quick Check:

    Most weight is on 30 so mean > 20 - 23 fits ✅

Hint: Frequencies act as weights; use Σ(f×x)/Σf.
Common Mistakes: Dividing by number of classes instead of total frequency.
3. Find the weighted standard deviation for values 2, 4, 6 with weights 1, 1, 2 respectively (round to 2 decimals).
medium
A. 1.50
B. 1.25
C. 1.80
D. 1.66

Solution

  1. Step 1: Compute weighted mean

    Σ(w×x) = (1×2)+(1×4)+(2×6)=2+4+12=18. Σw = 1+1+2=4.
    Weighted mean μ = 18 ÷ 4 = 4.5.

  2. Step 2: Compute weighted squared deviations

    For 2: (2-4.5)² = 6.25 → weighted = 1×6.25 = 6.25.
    For 4: (4-4.5)² = 0.25 → weighted = 1×0.25 = 0.25.
    For 6: (6-4.5)² = 2.25 → weighted = 2×2.25 = 4.50.
    Σw×(x-μ)² = 6.25 + 0.25 + 4.50 = 11.00.

  3. Step 3: Weighted variance & SD

    Variance = 11.00 ÷ Σw = 11.00 ÷ 4 = 2.75.
    SD = √2.75 ≈ 1.6583 ≈ 1.66.

  4. Final Answer:

    1.66 → Option D.

  5. Quick Check:

    SD small since values are close; heavier weight on 6 pulls mean up - calculation consistent ✅

Hint: Compute μ first, then Σ[w×(x-μ)²] ÷ Σw, then square-root.
Common Mistakes: Forgetting to multiply squared deviations by weights before summing.
4. An exam has three sections with weights 30%, 30%, 40%. Scores: 78, 82, 90. Find the weighted mean score.
medium
A. 84.00
B. 83.40
C. 82.60
D. 85.00

Solution

  1. Step 1: Convert percentage weights to decimals

    Weights: 0.30, 0.30, 0.40.

  2. Step 2: Compute weighted sum

    Weighted sum = 0.30×78 + 0.30×82 + 0.40×90 = 23.4 + 24.6 + 36 = 84.0.

  3. Step 3: Weighted mean

    Weighted mean = 84.0 (weights sum to 1).

  4. Final Answer:

    84.00 → Option A.

  5. Quick Check:

    Highest weight on 90 pulls mean upward; 84 is between 82 and 90 - sensible ✅

Hint: If weights sum to 1, the weighted mean is just Σ(w×x).
Common Mistakes: Forgetting to convert percentages to decimals or not ensuring weights sum to 1.
5. Values 10, 20, 30 have weights 2, 3, 5 respectively. Find the weighted standard deviation (round to 2 decimals).
medium
A. 7.55
B. 8.00
C. 7.81
D. 9.00

Solution

  1. Step 1: Compute weighted mean

    Σ(w×x) = (2×10)+(3×20)+(5×30)=20+60+150=230. Σw=2+3+5=10.
    Weighted mean μ = 230 ÷ 10 = 23.0.

  2. Step 2: Compute weighted squared deviations

    For 10: (10-23)² = 169 → weighted = 2×169 = 338.
    For 20: (20-23)² = 9 → weighted = 3×9 = 27.
    For 30: (30-23)² = 49 → weighted = 5×49 = 245.
    Σw×(x-μ)² = 338 + 27 + 245 = 610.

  3. Step 3: Weighted variance & SD

    Variance = 610 ÷ 10 = 61. SD = √61 ≈ 7.81.

  4. Final Answer:

    7.81 → Option C.

  5. Quick Check:

    Distribution spans 10-30; weighted mean 23 and SD ≈ 7.81 are consistent ✅

Hint: Do Σ(w×x) then Σ[w×(x-μ)²] ÷ Σw, then sqrt.
Common Mistakes: Using unweighted variance formula or dividing by number of unique values instead of Σw.

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