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Variance and Standard Deviation Formula (Direct Method)

Introduction

Variance and Standard Deviation (SD) are key measures of how much data values deviate from their mean. While the mean tells you the central value, the variance and standard deviation show how spread out the data is around that mean.

This pattern is important because it forms the foundation for almost every topic in data analysis, statistics, and aptitude questions on consistency, reliability, or risk.

Pattern: Variance and Standard Deviation Formula (Direct Method)

Pattern

The key concept: Variance measures the average squared deviation from the mean, and Standard Deviation is the square root of the variance.

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

Step-by-Step Example

Question

Find the variance and standard deviation for the numbers 5, 7, and 9.

Solution

  1. Step 1: Identify the given data

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

  2. Step 2: Calculate the mean (x̄)

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

  3. Step 3: Find each deviation (x - x̄) and its square

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

  4. Step 4: Apply the formula for variance

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

  5. Step 5: Find the standard deviation

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

  6. Final Answer:

    Variance = 2.67
    Standard Deviation = 1.63

  7. Quick Check:

    If all numbers are close to the mean (7), SD will be small (≈1.6) ✅

Quick Variations

1. For larger datasets, use the same formula with more observations.

2. If data has frequencies, multiply each (x - mean)² by its frequency before summing.

3. For equally spaced values, use pattern shortcuts to compute variance and SD faster.

Trick to Always Use

  • Step 1: Calculate the mean - it’s the base for all deviation calculations.
  • Step 2: Find each deviation (x - mean), square it, and sum them up.
  • Step 3: Divide the total of squared deviations by the number of items to get variance.
  • Step 4: Take the square root of variance to find the standard deviation.

Summary

Summary

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

  • Variance = average of squared deviations from the mean.
  • Standard Deviation = square root of the variance.
  • Smaller SD → data are closely packed; larger SD → data are more spread out.
  • Always calculate the mean first, then deviations.
  • In aptitude questions, a smaller SD indicates more consistency or stability in data.

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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