Standard Deviation

Standard Deviation (σ) is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It indicates how spread out the data points are from the mean.

1 Definition

The standard deviation is the square root of variance and is given by:

• Population Standard Deviation (σ):
  • σ = √[(1/n) * Σ (xᵢ - μ)²]
• Sample Standard Deviation (s):
  • s = √[(1/(n-1)) * Σ (xᵢ - x̄)²]

where:

• xᵢ – Each data point.
• μ – Population mean.
• x̄ – Sample mean.
• n – Number of data points.
• Σ – Summation notation.

2 Interpretation

• Low Standard Deviation – Data points are close to the mean.
• High Standard Deviation – Data points are widely spread.
• Zero Standard Deviation – All values are identical.

3 Example Calculation

Given dataset: [5, 7, 9, 10, 14]

1. Calculate the mean:
   • x̄ = (5 + 7 + 9 + 10 + 14) / 5 = 9
2. Compute squared differences:
   • (5 - 9)² = 16
   • (7 - 9)² = 4
   • (9 - 9)² = 0
   • (10 - 9)² = 1
   • (14 - 9)² = 25
3. Population variance:
   • σ² = (16 + 4 + 0 + 1 + 25) / 5 = 9.2
4. Population standard deviation:
   • σ = √9.2 ≈ 3.03
5. Sample standard deviation:
   • s² = (16 + 4 + 0 + 1 + 25) / (5-1) = 11.5
   • s = √11.5 ≈ 3.39

4 Properties

• Always non-negative – Standard deviation is always ≥ 0.
• Same units as the data – Unlike variance, which is squared.
• Sensitive to outliers – Extreme values can significantly impact standard deviation.

5 Applications

• Finance – Measures risk (volatility of asset returns).
• Machine Learning – Evaluates feature distribution and normalization.
• Quality Control – Assesses consistency in production processes.

6 Standard Deviation vs. Variance

Measure	Formula	Interpretation
Variance (σ²)	(1/n) * Σ (xᵢ - μ)²	Average squared deviation from mean.
Standard Deviation (σ)	√Variance	Dispersion in original units.

7 See Also

• Variance
• Mean
• Normal Distribution
• Statistical Dispersion
• Coefficient of Variation