Does zero correlation mean independence?
No, zero correlation does not mean independence. When r = 0, the two variables are uncorrelated, meaning there is no linear relation between them. However, this does not imply that the variables are independent of each other.
The context clearly states that when r = 0, there is no linear relation between the variables, but other types of relation may be there. A low value of r (close to zero) indicates a weak linear relation, but there may be a non-linear relation between the variables.
Therefore, zero correlation only indicates the absence of a linear relationship, not complete independence.
Explanation
The textbook explicitly states that when r = 0, the two variables are uncorrelated and there is no linear relation between them. However, it clarifies that 'other types of relation may be there.' This is a crucial distinction in statistics - correlation measures only linear relationships. Variables can have strong non-linear relationships (like a perfect U-shaped pattern) while still having zero correlation. Independence is a stronger condition that requires no relationship of any kind between variables. The textbook emphasizes this by noting that a low value of r indicates weak linear relation 'But there may be a non-linear relation.'