Standard Deviation and Variance: Measuring Data Spread and Bell Curves
Learn how standard deviation quantifies volatility and consistency across exam scores, investments, and experiments.
Why the Average (Mean) Isn't Enough
Consider two investment funds that both produced an average annual return of 8% over five years:
• Fund A annual returns: 7%, 8%, 9%, 8%, 8% (consistent, low risk)
• Fund B annual returns: −15%, 32%, −8%, 24%, 7% (volatile, wild swings)
Both funds have identical means (8%), but their risk profiles are completely different. Standard deviation tells us how tightly or widely values are scattered around the mean.
Step-by-Step Calculation Guide
1. Calculate the mean (x̄): (4 + 8 + 6 + 5 + 7) ÷ 5 = 6.
2. Find deviations from mean:
(4 − 6) = −2 → (−2)² = 4
(8 − 6) = +2 → (+2)² = 4
(6 − 6) = 0 → (0)² = 0
(5 − 6) = −1 → (−1)² = 1
(7 − 6) = +1 → (+1)² = 1
3. Sum squared deviations: 4 + 4 + 0 + 1 + 1 = 10.
4. Sample Variance (divide by n − 1 = 4): 10 ÷ 4 = 2.5.
5. Standard Deviation (square root): √2.5 ≈ 1.58.
The Empirical Rule (68-95-99.7 Rule)
For normally distributed (bell-shaped) data:
• 68.2% of all data points fall within 1 standard deviation of the mean (±1σ).
• 95.4% fall within 2 standard deviations (±2σ).
• 99.7% fall within 3 standard deviations (±3σ).
Values beyond 3 standard deviations are statistically rare and are typically classified as outliers.