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Variance and Standard Deviation Formula (Direct Method)

Introduction

Variance और Standard Deviation (SD) ऐसे key measures हैं जो बताते हैं कि data values अपने mean से कितना deviate करती हैं। जहाँ mean आपको central value बताता है, वहीं variance और standard deviation यह दिखाते हैं कि data mean के आसपास कितना फैला हुआ है।

यह pattern important है क्योंकि यह data analysis, statistics और aptitude questions जैसे consistency, reliability या risk वाले topics की foundation बनाता है।

Pattern: Variance and Standard Deviation Formula (Direct Method)

Pattern

मुख्य concept: Variance औसत squared deviation को measure करता है, और Standard Deviation variance का square root होता है।

Formulas:
Variance (σ²) = [ Σ (x - x̄)² ] ÷ n
Standard Deviation (σ) = √(Variance) = √([ Σ (x - x̄)² ] ÷ n)

Step-by-Step Example

Question

5, 7 और 9 के लिए variance और standard deviation निकालें।

Solution

  1. Step 1: दिए गए डेटा की पहचान करें

    Data: 5, 7, 9
    Number of items (n) = 3

  2. Step 2: Mean (x̄) निकालें

    Mean (x̄) = (5 + 7 + 9) ÷ 3 = 21 ÷ 3 = 7

  3. Step 3: हर deviation (x - x̄) और उसका square निकालें

    Deviation और Square Calculation
    Value (x)Deviation (x - 7)(x - 7)²
    5-24
    700
    9+24

  4. Step 4: Variance formula apply करें

    Variance (σ²) = (4 + 0 + 4) ÷ 3 = 8 ÷ 3 = 2.67

  5. Step 5: Standard Deviation निकालें

    Standard Deviation (σ) = √(2.67) ≈ 1.63

  6. Final Answer:

    Variance = 2.67
    Standard Deviation = 1.63

  7. Quick Check:

    अगर सभी numbers mean (7) के क़रीब हैं, तो SD छोटा होगा (≈1.6) ✅

Quick Variations

1. बड़े datasets के लिए भी यही formula ज़्यादा observations के साथ apply होता है।

2. अगर data में frequencies हों, तो हर (x - mean)² को उसकी frequency से multiply करके फिर sum करें।

3. Equally spaced values के लिए shortcuts से variance और SD जल्दी निकाले जा सकते हैं।

Trick to Always Use

  • Step 1: Mean निकालें - यही सभी deviation calculations का base है।
  • Step 2: हर deviation (x - mean) निकालें, उसे square करें और total बनाएं।
  • Step 3: Total squared deviations को number of items से divide करें - यही variance है।
  • Step 4: Variance का square root लें - यही standard deviation है।

Summary

Summary

In the Variance and Standard Deviation (Direct Method) pattern:

  • Variance = mean से squared deviations का average।
  • Standard Deviation = variance का square root।
  • छोटा SD → data ज़्यादा करीब-करीब; बड़ा SD → data ज़्यादा फैला हुआ।
  • हमेशा पहले mean निकालें, फिर deviations।
  • Aptitude questions में छोटा SD ज़्यादा consistency दिखाता है।

Practice

(1/5)
1. Find the variance and standard deviation for the numbers 2, 4, and 6.
easy
A. Variance=2.67, SD=1.63
B. Variance=3, SD=1.73
C. Variance=4, SD=2
D. Variance=2, SD=1.41

Solution

  1. Step 1: Identify the data

    Numbers = 2, 4, 6; n = 3.

  2. Step 2: Find the mean

    Mean = (2 + 4 + 6) ÷ 3 = 12 ÷ 3 = 4.

  3. Step 3: Compute deviations and squares

    Deviations: 2-4=-2, 4-4=0, 6-4=+2. Squares: 4, 0, 4.

  4. Step 4: Calculate variance and SD

    Variance = (4 + 0 + 4) ÷ 3 = 8 ÷ 3 = 2.67; SD = √2.67 ≈ 1.63.

  5. Final Answer:

    Variance = 2.67, SD = 1.63 → Option A.

  6. Quick Check:

    Sum of squared deviations = 8; dividing by 3 gives 2.67 ✅
Hint: Find mean first, then average of squared deviations.
Common Mistakes: Taking square root before dividing by n.
2. The numbers are 5, 7, and 9. Find their variance and standard deviation.
easy
A. Variance=3, SD=1.73
B. Variance=2.67, SD=1.63
C. Variance=2, SD=1.41
D. Variance=4, SD=2

Solution

  1. Step 1: Identify data

    Values: 5, 7, 9; n = 3.

  2. Step 2: Find mean

    Mean = (5 + 7 + 9) ÷ 3 = 21 ÷ 3 = 7.

  3. Step 3: Compute deviations and squares

    Deviations: 5-7=-2, 7-7=0, 9-7=+2 → Squares: 4, 0, 4.

  4. Step 4: Calculate variance and SD

    Variance = (4 + 0 + 4) ÷ 3 = 8 ÷ 3 = 2.67; SD = √2.67 ≈ 1.63.

  5. Final Answer:

    Variance = 2.67, SD = 1.63 → Option B.

  6. Quick Check:

    Pattern matches any shift of an evenly spaced trio; variance unchanged ✅
Hint: For equally spaced triplets, shifting all values doesn't change variance.
Common Mistakes: Forgetting to square negative deviations.
3. For the data 10, 20, and 30, find variance and standard deviation.
easy
A. Variance=66.67, SD=8.16
B. Variance=100, SD=10
C. Variance=50, SD=7.07
D. Variance=80, SD=9

Solution

  1. Step 1: Identify data

    Values: 10, 20, 30; n = 3.

  2. Step 2: Find mean

    Mean = (10 + 20 + 30) ÷ 3 = 60 ÷ 3 = 20.

  3. Step 3: Calculate deviations and squares

    Deviations: 10-20=-10; 20-20=0; 30-20=+10 → Squares: 100, 0, 100.

  4. Step 4: Apply formula

    Variance = (100 + 0 + 100) ÷ 3 = 200 ÷ 3 = 66.67; SD = √66.67 ≈ 8.16.

  5. Final Answer:

    Variance = 66.67, SD = 8.16 → Option A.

  6. Quick Check:

    Large gaps increase variance; 200 ÷ 3 = 66.67 ✅
Hint: Bigger gaps in data increase variance sharply.
Common Mistakes: Using (n-1) instead of n in denominator for population SD.
4. For 4, 6, 8, and 10, find the variance and standard deviation.
medium
A. Variance=4, SD=2
B. Variance=3.5, SD=1.87
C. Variance=5, SD=2.24
D. Variance=6, SD=2.45

Solution

  1. Step 1: Identify data

    Values = 4, 6, 8, 10; n = 4.

  2. Step 2: Find mean

    Mean = (4 + 6 + 8 + 10) ÷ 4 = 28 ÷ 4 = 7.

  3. Step 3: Compute deviations and squares

    Deviations: 4-7=-3; 6-7=-1; 8-7=+1; 10-7=+3 → Squares: 9, 1, 1, 9.

  4. Step 4: Find variance and SD

    Variance = (9 + 1 + 1 + 9) ÷ 4 = 20 ÷ 4 = 5; SD = √5 ≈ 2.24.

  5. Final Answer:

    Variance = 5, SD = 2.24 → Option C.

  6. Quick Check:

    Sum of squared deviations = 20; 20 ÷ 4 = 5 ✅
Hint: Even spacing with more numbers increases precision of mean and SD.
Common Mistakes: Adding squared deviations incorrectly.
5. For the numbers 3, 9, 12, and 18, find the variance and standard deviation.
medium
A. Variance=36.75, SD=6.06
B. Variance=35, SD=5.92
C. Variance=30, SD=5.48
D. Variance=29.25, SD=5.41

Solution

  1. Step 1: Identify given data

    Numbers = 3, 9, 12, 18; n = 4.

  2. Step 2: Find mean

    Mean = (3 + 9 + 12 + 18) ÷ 4 = 42 ÷ 4 = 10.5.

  3. Step 3: Compute deviations and squares

    Deviations: 3-10.5=-7.5; 9-10.5=-1.5; 12-10.5=+1.5; 18-10.5=+7.5 → Squares: 56.25, 2.25, 2.25, 56.25.

  4. Step 4: Calculate variance and SD

    Variance = (56.25 + 2.25 + 2.25 + 56.25) ÷ 4 = 117 ÷ 4 = 29.25; SD = √29.25 ≈ 5.41.

  5. Final Answer:

    Variance = 29.25, SD = 5.41 → Option D.

  6. Quick Check:

    Sum of squared deviations = 117; 117 ÷ 4 = 29.25 ✅
Hint: Use careful decimal handling for means and squares; keep two decimals for SD.
Common Mistakes: Squaring deviations incorrectly or using wrong denominator (n-1).

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