If we select a random sample of data for two variables and, after computing the correlation coefficient,

If we select a random sample of data for two variables and, after computing the correlation coefficient, conclude that the two variables may have zero correlation, can we say that there is no relationship between the two variables? Discuss.

Answer 1: If we take a random sample of data for two variables, and after computing, there is zero correlation, this only tells us that there is no linear relationship between the two. But they might have other forms of relationship between them, like cubic or trigonometric or quadratic. So we cannot with all confidence say they have no relationship between them if the correlation is zero.

Answer 2: If the correlation coefficient of two variables is zero, then it does not necessarily imply that there is no relationship between them. The correlation coefficient measures the degree of linear association between the two variables. If the underlying relationship between the two variables is anything other than linear (for example, exponential), then it will not be captured by the correlation coefficient. In such a case, the correlation coefficient will be zero, when actually there is non linear relationship between the two variables.