gen-define-variance-and-covariance-and-give-one-short-worked-exa-60f5

Define variance and covariance and give one short worked example.

# Variance and Covariance ## Definitions **Variance** measures how far a set of numbers spread out from their average value. - Formula: **Var(X) = E[(X - μ)²]** where μ is the mean - Alternative: **Var(X) = E[X²] - (E[X])²** **Covariance** measures how two variables change together. - Formula: **Cov(X,Y) = E[(X - μₓ)(Y - μᵧ)]** - Alternative: **Cov(X,Y) = E[XY] - E[X]E[Y]** ## Worked Example Given data for study hours (X) and test scores (Y): | X (hours) | 2 | 4 | 6 | |-----------|---|---|---| | Y (score) | 60 | 75 | 90 | **Find Var(X) and Cov(X,Y):** **Step 1:** Calculate means - E[X] = (2 + 4 + 6)/3 = 4 - E[Y] = (60 + 75 + 90)/3 = 75 **Step 2:** Variance of X - Var(X) = [(2-4)² + (4-4)² + (6-4)²]/3 - Var(X) = [4 + 0 + 4]/3 = **8/3 ≈ 2.67** **Step 3:** Covariance of X and Y - Cov(X,Y) = [(2-4)(60-75) + (4-4)(75-75) + (6-4)(90-75)]/3 - Cov(X,Y) = [(-2)(-15) + 0 + (2)(15)]/3 - Cov(X,Y) = [30 + 0 + 30]/3 = **20** The positive covariance indicates that study hours and test scores tend to increase together.