How to Calculate Covariance
What is Covariance?
Covariance measures how two variables change together. Positive covariance means they tend to increase together; negative means one increases as the other decreases. It is the foundation of correlation and portfolio theory.
Formula
Cov(X,Y) = E[(X − μₓ)(Y − μᵧ)] = (Σ(xᵢ − x̄)(yᵢ − ȳ)) / (n−1)
- X, Y
- two variables/datasets
- μₓ, μᵧ
- means of X and Y
- Cov(X,Y)
- covariance — measure of joint variability
- n
- number of data points
Step-by-Step Guide
- 1Sample covariance: Cov(X,Y) = Σ(xᵢ−x̄)(yᵢ−ȳ) / (n−1)
- 2Positive: variables move together
- 3Negative: variables move oppositely
- 4Zero: no linear relationship
Worked Examples
Input
X=[2,4,4,4,5], Y=[1,3,3,4,4]
Result
Cov ≈ 0.95 (positive relationship)
Frequently Asked Questions
What does positive covariance mean?
Positive covariance: when X increases, Y tends to increase. They vary together.
What does zero covariance mean?
Zero covariance suggests no linear relationship, but nonlinear relationships may exist.
How is covariance related to correlation?
Correlation = Cov(X,Y) / (σₓ × σᵧ). Correlation is covariance normalized to [−1, 1].
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