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.

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.

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.

Example Calculation

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

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

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.

Applications

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

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.