What is Variance?
Variance measures how far a set of numbers is spread out from their average value. It's the average of the squared differences from the mean—and it's the foundation upon which standard deviation is built.
Each bar shows squared deviation from mean. Variance = average of these bars.
Variance Formula
Population Variance
σ² = Σ(xᵢ - μ)² / N
Sample Variance
s² = Σ(xᵢ - x̄)² / (n-1)
1
Calculate the mean
Add all values and divide by the count.
2
Find each deviation
Subtract the mean from each data point.
3
Square each deviation
This eliminates negative values and emphasizes large deviations.
4
Average the squared deviations
Divide by N (population) or n-1 (sample).
Why Do We Square Deviations?
Three Key Reasons
1. Eliminate negatives: Without squaring, positive and negative deviations would cancel out, making the sum zero.
2. Penalize outliers: Squaring gives more weight to values far from the mean.
3. Mathematical properties: Variance has useful algebraic properties for statistical inference.
Example: Why Not Just Use Absolute Values?
Dataset: 2, 4, 4, 4, 5, 5, 7, 9 (Mean = 5)
Mean Absolute Deviation:
|2-5| + |4-5| + ... = 14
MAD = 14/8 = 1.75
Variance (squared):
(2-5)² + (4-5)² + ... = 32
Var = 32/8 = 4
Variance vs Standard Deviation
The Relationship
Standard Deviation = √Variance → σ = √σ²
Variance (σ²)
- Units are squared (e.g., cm², $²)
- Harder to interpret directly
- Useful for mathematical operations
- Additive for independent variables
Standard Deviation (σ)
- Same units as original data
- Easier to interpret
- Better for communication
- Used in z-scores and confidence intervals
Applications of Variance
While standard deviation is more commonly reported, variance has specific uses:
- ANOVA:Analysis of Variance compares means across groups
- Portfolio Theory:Variances of returns are used in optimization
- Regression:R² is explained variance divided by total variance
- PCA:Principal Component Analysis maximizes explained variance