Variance

Variance is a statistical measure that quantifies the dispersion of a dataset relative to its mean. It indicates how much the values in a dataset deviate from the average value.

Definition

The variance (σ²) of a dataset with n elements is calculated as:

Population Variance (σ²):
- σ² = (1/n) * Σ (xᵢ - μ)²
Sample Variance (s²):
- s² = (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 Variance – Data points are close to the mean.
High Variance – Data points are spread out from the mean.
Zero Variance – 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
Sample variance:
- s² = (16 + 4 + 0 + 1 + 25) / (5-1) = 11.5

Properties

Always non-negative – Squaring deviations ensures variance is ≥ 0.
Measured in squared units – Different from the original data scale.
Used to compute standard deviation – Standard deviation (σ) is the square root of variance.

Applications

Finance – Measuring risk (volatility of asset returns).
Machine Learning – Evaluating feature distribution in datasets.
Quality Control – Assessing variability in production processes.

Variance vs. Standard Deviation

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