Explanation

Correlation measures the linear relationship between two variables. A zero correlation indicates that there is no linear relationship between the variables.

Key Points:

1. Correlation (ρ) = 0 means the variables have zero linear dependence.
2. However, this does not necessarily mean independence - the variables could still have a non-linear relationship (e.g., quadratic, parabolic, or other functional forms).
3. Examples given in the text:
   • A symmetric parabola: Corr(X, Y) = 0
   • A symmetric V-shape graph: Corr(X, Y) = 0 Both show non-linear relationships despite zero correlation.

Why Other Options Are Incorrect:

• A: Incorrect - Zero correlation does not guarantee independence; variables can be dependent in non-linear ways.
• C: Incorrect - Zero correlation indicates the absence of linear dependence, not linear dependence.
• D: Incorrect - A negative linear relationship would have correlation < 0, not = 0.

Statistical Insight:

Correlation coefficient (ρ) ranges from -1 to +1:

• ρ = 0: No linear relationship
• ρ > 0: Positive linear relationship
• ρ < 0: Negative linear relationship

Zero correlation only eliminates linear patterns, leaving the possibility of other functional relationships intact.