Covariance Formula
Measure how two variables change together.
Essential for portfolio analysis, regression, and data science.
The Formula
Cov(X, Y) = Σ (xᵢ - x̄)(yᵢ - ȳ) / (n - 1)
Covariance measures the direction of the relationship between two variables. A positive value means they tend to increase together. A negative value means one increases as the other decreases.
Variables
| Symbol | Meaning |
|---|---|
| Cov(X, Y) | Covariance between variables X and Y |
| xᵢ, yᵢ | Individual data points |
| x̄, ȳ | Mean (average) of X and Y |
| n | Number of data points |
Example 1
Find covariance for X = {1, 2, 3} and Y = {2, 4, 5}
x̄ = (1+2+3)/3 = 2, ȳ = (2+4+5)/3 = 3.67
Σ(xᵢ - x̄)(yᵢ - ȳ) = (1-2)(2-3.67) + (2-2)(4-3.67) + (3-2)(5-3.67)
= (-1)(-1.67) + (0)(0.33) + (1)(1.33) = 1.67 + 0 + 1.33 = 3.0
Cov(X,Y) = 3.0 / (3-1) = 1.5 (positive — they increase together)
Interpreting Covariance
- Cov > 0: Variables tend to move in the same direction
- Cov < 0: Variables tend to move in opposite directions
- Cov ≈ 0: No clear linear relationship
When to Use It
Use covariance when:
- Building an investment portfolio (diversification analysis)
- Performing linear regression analysis
- Exploring relationships between variables in data science
- Calculating the correlation coefficient (which normalizes covariance)