True or false, and explain: if a correlation coefficient between two variables is zero, there is no relat?

False.





The correlation coefficient only measures linear relationships - that is, relationships between two variables (call them x and y) that can be written in the form y = mx + c. It is a measure of how well the data points fall on a straight line.





But not all relationships are linear. What about quadratic relationships? The points will not fall on a straight line, so the correlation coefficient will be zero (or very close to) but there is still a relationship between the variables, the data points would fit a parabola (y = k(x - a)^2 + c). And what about other relationships? Trig functions, exponential functions etc. They're not linear, so the correlation coefficient will be close to zero, but there is still a clear relationship between the two variables.