Standard Deviation Calculator – Variance & SD Online

Example 1: Small dataset

Problem: Find standard deviation of {2, 4, 6, 8, 10}

Step 1: Mean = (2+4+6+8+10)/5 = 30/5 = 6

Step 2: Squared differences: (2-6)²=16, (4-6)²=4, (6-6)²=0, (8-6)²=4, (10-6)²=16

Step 3: Sum = 16+4+0+4+16 = 40

Step 4: Population variance = 40/5 = 8

Step 5: Standard deviation = √8 ≈ 2.83

Example 2: Sample vs Population

Problem: Same data {2, 4, 6, 8, 10}, but treat as a sample

Steps 1-3 are identical. Sum of squared differences = 40

Step 4: Sample variance = 40/(5-1) = 40/4 = 10

Step 5: Sample standard deviation = √10 ≈ 3.16

Notice: Sample SD (3.16) is larger than population SD (2.83) – the n-1 correction increases the estimate.

Example 3: Interpreting results

Problem: Class A has mean=75, SD=5. Class B has mean=75, SD=15. What does this mean?

Solution: Both classes average 75%, but Class A is more consistent.

Class A: Most scores between 70-80 (within 1 SD)

Class B: Scores spread from 60-90 (within 1 SD). More variability in performance.

Example 4: Coefficient of Variation

Problem: Dataset A: mean=100, SD=10. Dataset B: mean=50, SD=8. Which is more variable?

Solution: Compare coefficient of variation (CV = SD/mean × 100%)

CV(A) = 10/100 × 100% = 10%

CV(B) = 8/50 × 100% = 16%. Dataset B is relatively more variable despite smaller SD.

Example 5: Real-world application

Problem: Investment A returns 8% ± 2%. Investment B returns 10% ± 8%. Which is safer?

Solution: Standard deviation measures risk/volatility.

Investment A: Lower return but more predictable (SD=2%)

Investment B: Higher return but riskier (SD=8%). Returns could range from 2% to 18%.