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Combined Standard Deviation (Two Data Sets)

Introduction

कई बार दो अलग-अलग groups के observations को एक साथ जोड़कर उनका combined standard deviation निकालना पड़ता है। Combined SD में दोनों groups के size, mean और variance का effect शामिल होता है - इसमें group के अंदर का spread और group means के अंतर दोनों को consider किया जाता है।

यह pattern तब useful होता है जब दो class scores, दो batches के measurements या दो subgroups के sample results को merge करना हो।

Pattern: Combined Standard Deviation (Two Data Sets)

Pattern

मुख्य concept: Combined variance = हर group के (variance + उसके mean और combined mean के अंतर का square) का weighted sum, जिसे total size से divide किया जाता है। Combined SD = इसी variance का square root।

Group 1: size n₁, mean x̄₁, SD σ₁.
Group 2: size n₂, mean x̄₂, SD σ₂.

Combined mean:
x̄ = (n₁ × x̄₁ + n₂ × x̄₂) ÷ (n₁ + n₂)

Combined variance formula:
σ² = [ n₁ × (σ₁² + (x̄₁ - x̄)²) + n₂ × (σ₂² + (x̄₂ - x̄)²) ] ÷ (n₁ + n₂)
Combined standard deviation:
σ = √σ²

Step-by-Step Example

Question

Class A: n₁ = 10 students, mean = 50, SD = 4.
Class B: n₂ = 15 students, mean = 55, SD = 5.
सभी 25 students के लिए combined mean और combined SD निकालें।

Solution

  1. Step 1: Combined mean x̄ निकालें

    x̄ = (n₁ × x̄₁ + n₂ × x̄₂) ÷ (n₁ + n₂)
    = (10 × 50 + 15 × 55) ÷ 25
    = (500 + 825) ÷ 25 = 1,325 ÷ 25 = 53

  2. Step 2: Group means के combined mean से deviations निकालें

    (x̄₁ - x̄) = 50 - 53 = -3 → (x̄₁ - x̄)² = 9
    (x̄₂ - x̄) = 55 - 53 = 2 → (x̄₂ - x̄)² = 4

  3. Step 3: n × (σ² + (x̄ - x̄)²) दोनों groups के लिए निकालें

    Group A: n₁ × (σ₁² + (x̄₁ - x̄)²) = 10 × (4² + 9) = 10 × (16 + 9) = 10 × 25 = 250
    Group B: n₂ × (σ₂² + (x̄₂ - x̄)²) = 15 × (5² + 4) = 15 × (25 + 4) = 15 × 29 = 435

  4. Step 4: Sum करें और total size से divide करके combined variance निकालें

    Total = 250 + 435 = 685
    Combined variance σ² = 685 ÷ 25 = 27.4

  5. Step 5: Combined SD निकालें

    σ = √27.4 ≈ 5.24

  6. Final Answer:

    Combined Mean = 53
    Combined SD ≈ 5.24

  7. Quick Check:

    Combined mean (53) दोनों means के बीच है और Class B के mean के करीब है क्योंकि उसकी strength ज़्यादा है।
    Combined SD (≈5.24) दोनों SDs (4 और 5) के बीच है - mean difference की वजह से थोड़ा बढ़ा हुआ है। ✅

Quick Variations

1. अगर दोनों groups के size equal हों (n₁ = n₂), तो combined mean = दोनों means का average।

2. अगर दोनों groups के mean equal हों (x̄₁ = x̄₂), तो combined variance simplify होकर बनता है: [n₁σ₁² + n₂σ₂²] ÷ (n₁ + n₂)।

3. दो से ज़्यादा groups हों तो formula को extend करके सभी groups के nᵢ(σᵢ² + (x̄ᵢ - x̄)²) का sum लें और total N से divide करें।

Trick to Always Use

  • Step 1: सबसे पहले combined mean निकालें - यह आगे की calculation के लिए जरूरी है।
  • Step 2: Formula के bracket में दो parts होते हैं: group का internal variance + group mean और combined mean का अंतर।
  • Step 3: बड़ा group (higher n) combined SD पर ज़्यादा effect डालता है।

Summary

Summary

In the Combined Standard Deviation (Two Data Sets) pattern:

  • Combined Mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂)
  • Combined Variance = [ n₁(σ₁² + (x̄₁ - x̄)²) + n₂(σ₂² + (x̄₂ - x̄)²) ] ÷ (n₁ + n₂)
  • Combined SD = √(Combined Variance)
  • Formula दोनों groups के अंदर की variability और उनके means के अंतर दोनों को consider करता है।

Practice

(1/5)
1. Class A has 20 students with Mean = 60 and SD = 4. Class B has 30 students with Mean = 65 and SD = 5. Find the combined Mean of both classes.
easy
A. 63.0
B. 62.5
C. 64.0
D. 61.5

Solution

  1. Step 1: Identify values

    n₁ = 20, x̄₁ = 60; n₂ = 30, x̄₂ = 65.

  2. Step 2: Apply combined mean formula

    x̄ = (n₁×x̄₁ + n₂×x̄₂) ÷ (n₁ + n₂) = (20×60 + 30×65) ÷ 50 = (1,200 + 1,950) ÷ 50 = 3,150 ÷ 50 = 63.0.

  3. Final Answer:

    Combined Mean = 63.0 → Option A.

  4. Quick Check:

    63 lies between 60 and 65 and is closer to 65 (larger group) - consistent ✅

Hint: Use weighted average: larger group pulls the combined mean toward its mean.
Common Mistakes: Taking simple average of the two means instead of weighting by group sizes.
2. Two groups have means 40 and 50 with equal sizes of 10 each. Find the combined mean.
easy
A. 44
B. 45
C. 48
D. 46

Solution

  1. Step 1: Note equal sizes

    When group sizes are equal, combined mean = (mean₁ + mean₂) ÷ 2.

  2. Step 2: Compute

    (40 + 50) ÷ 2 = 45.

  3. Final Answer:

    Combined Mean = 45 → Option B.

  4. Quick Check:

    45 is midway between 40 and 50 ✅

Hint: If group sizes equal, take simple average of means.
Common Mistakes: Applying weights when sizes are equal (unnecessary).
3. Group A: n₁ = 10, Mean = 70, SD = 3; Group B: n₂ = 20, Mean = 75, SD = 4. Find the combined Mean.
easy
A. 72.5
B. 74.0
C. 73.33
D. 76.0

Solution

  1. Step 1: Apply weighted mean formula

    x̄ = (n₁×x̄₁ + n₂×x̄₂) ÷ (n₁ + n₂) = (10×70 + 20×75) ÷ 30.

  2. Step 2: Compute

    (700 + 1,500) ÷ 30 = 2,200 ÷ 30 = 73.33 (rounded to 2 decimals).

  3. Final Answer:

    Combined Mean ≈ 73.33 → Option C.

  4. Quick Check:

    Value lies between 70 and 75 and is closer to 75 (larger group) ✅

Hint: Larger group’s mean has greater influence on combined mean.
Common Mistakes: Using simple average instead of weighted average.
4. Class X: n₁ = 12, Mean = 50, SD = 3. Class Y: n₂ = 18, Mean = 55, SD = 4. Find the combined Standard Deviation (rounded to 2 decimals).
medium
A. 3.80
B. 4.00
C. 4.50
D. 4.38

Solution

  1. Step 1: Compute combined mean

    x̄ = (12×50 + 18×55) ÷ 30 = (600 + 990) ÷ 30 = 1,590 ÷ 30 = 53.

  2. Step 2: Compute group contributions

    For X: σ₁² = 3² = 9; (x̄₁ - x̄)² = (50 - 53)² = 9 → term = 12×(9 + 9) = 12×18 = 216.
    For Y: σ₂² = 4² = 16; (x̄₂ - x̄)² = (55 - 53)² = 4 → term = 18×(16 + 4) = 18×20 = 360.

  3. Step 3: Combined variance and SD

    Sum = 216 + 360 = 576. Combined variance = 576 ÷ 30 = 19.2. Combined SD = √19.2 ≈ 4.38.

  4. Final Answer:

    Combined SD ≈ 4.38 → Option D.

  5. Quick Check:

    Combined SD ≈ 4.38 lies between 4 and 5 and is increased by the between-group difference - reasonable ✅

Hint: Include both within-group variance and squared mean differences when combining.
Common Mistakes: Ignoring the between-group term (x̄ᵢ - x̄)².
5. Two departments have the following data: Dept A: n₁ = 25, Mean = 80, SD = 5; Dept B: n₂ = 15, Mean = 70, SD = 4. Find the combined SD (rounded to 2 decimals).
medium
A. 6.72
B. 5.25
C. 6.00
D. 5.50

Solution

  1. Step 1: Combined mean

    x̄ = (25×80 + 15×70) ÷ 40 = (2,000 + 1,050) ÷ 40 = 3,050 ÷ 40 = 76.25.

  2. Step 2: Compute group contributions

    Dept A: σ₁² = 25; (x̄₁ - x̄)² = (80 - 76.25)² = 3.75² = 14.0625 → term = 25×(25 + 14.0625) = 25×39.0625 = 976.5625.
    Dept B: σ₂² = 16; (x̄₂ - x̄)² = (70 - 76.25)² = (-6.25)² = 39.0625 → term = 15×(16 + 39.0625) = 15×55.0625 = 825.9375.

  3. Step 3: Combined variance and SD

    Sum = 976.5625 + 825.9375 = 1,802.5. Combined variance = 1,802.5 ÷ 40 = 45.0625. Combined SD = √45.0625 ≈ 6.72.

  4. Final Answer:

    Combined SD ≈ 6.72 → Option A.

  5. Quick Check:

    Result is closer to the larger group’s SD but increased due to mean difference - makes sense ✅

Hint: Compute weighted sum of (variance + squared mean-difference) then divide by total N and square-root.
Common Mistakes: Forgetting to multiply each group's term by its size before summing.

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