Explanation
Correlation measures the linear relationship between two variables. A zero correlation indicates that there is no linear relationship between the variables.
Key Points:
- Correlation (ρ) = 0 means the variables have zero linear dependence.
- However, this does not necessarily mean independence - the variables could still have a non-linear relationship (e.g., quadratic, parabolic, or other functional forms).
- 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.
