Explanation

A zero correlation (correlation coefficient = 0) specifically indicates the absence of a linear relationship between two variables. However, it does NOT necessarily imply the absence of a non-linear relationship.

Key Points:

1. Correlation measures linear association - The correlation coefficient (Pearson's r) quantifies the strength and direction of a linear relationship between two variables.
2. Zero correlation means no linear pattern - When r = 0, there is no straight-line relationship between the variables.
3. Non-linear relationships can exist - Variables can have strong non-linear relationships (e.g., quadratic, exponential, sinusoidal) while still having zero linear correlation.

Example:

Consider the relationship y = x². If x ranges symmetrically around zero (e.g., -3, -2, -1, 0, 1, 2, 3), the correlation between x and y will be approximately zero, even though there is a perfect quadratic relationship.

Why other options are incorrect:

• Option B: Incorrect because zero correlation doesn't rule out non-linear relationships.
• Option C: Incorrect because zero correlation only rules out linear relationships, not necessarily non-linear ones.

Therefore, the correct answer is A - a zero correlation implies the absence of a linear relationship only.