Standard Deviation, Variance & Portfolio Volatility: Population (σ) vs Sample (s), The 68-95-99.7 Rule & Sharpe Ratio
In statistics, data science, and modern portfolio theory, the mean (average) only tells half the story. Variance ($\sigma^2$ or $s^2$) and Standard Deviation ($\sigma$ or $s$) measure data dispersion—quantifying how widely observations deviate from the center. In financial markets, standard deviation serves as the definitive mathematical proxy for investment volatility and downside risk.
format_list_bulleted Table of Contents
- 1. Population (σ) vs Sample (s) Formulas & Bessel's Correction ($n-1$)
- 2. The 68-95-99.7 Empirical Rule for Normal Distributions
- 3. Portfolio Risk & The Sharpe Ratio Formula
- 4. Step-by-Step Manual Calculation of Sample Standard Deviation
- 5. Real-World Applications: Quality Control (Six Sigma) & Polling
1. Population (σ) vs Sample (s) Formulas
When calculating standard deviation for a complete population vs a representative sample, mathematicians use Bessel's Correction ($n-1$) in the denominator to correct for sample bias:
σ = √ [ ∑ (xi − μ)2 ÷ N ]
Sample Standard Deviation (s — Bessel's Correction):
s = √ [ ∑ (xi − x̄)2 ÷ (n − 1) ]
2. The 68-95-99.7 Empirical Rule
In any symmetrical, Gaussian normal distribution:
- μ ± 1σ: Encompasses 68.27% of all outcomes.
- μ ± 2σ: Encompasses 95.45% of all outcomes (Threshold for 95% confidence intervals).
- μ ± 3σ: Encompasses 99.73% of all outcomes (Only 3 in 1,000 events are statistical anomalies).
3. Portfolio Risk & The Sharpe Ratio
In finance, standard deviation ($\sigma_p$) represents portfolio volatility. Nobel laureate William Sharpe created the Sharpe Ratio to measure excess return generated per unit of total risk:
A Sharpe Ratio > 1.0 is considered good, > 2.0 very good, and > 3.0 exceptional (institutional hedge fund quality).
4. Step-by-Step Manual Calculation (Dataset: 10, 12, 23, 23, 16, 23, 21, 16)
- Calculate Sample Mean (x̄): (10+12+23+23+16+23+21+16) ÷ 8 = 18.0
- Calculate Deviations & Squares: ∑ (xi − 18)2 = 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192
- Sample Variance (s2): 192 ÷ (8 − 1) = 27.43
- Sample Standard Deviation (s): √27.43 = 5.24.
5. Quality Control: Six Sigma ($6\sigma$)
In manufacturing and semiconductor fabrication, Motorola's Six Sigma standard demands that product tolerances be set at ±6 standard deviations from the mean, permitting no more than 3.4 defective parts per million opportunities (DPMO).