If coefficient of correlation, “r” between two variables is zero, does it mean that there is no relationship between the variables? Justify your answer”.

Solution:

A correlation coefficient of 0 means that there is no linear relationship between the variables. A correlation coefficient is a number between -1 and 1 which measures the degree to which two variables are linearly related. If there is perfect linear relationship with positive slope between the two variables, we have a correlation coefficient of 1; if there is positive correlation, whenever one variable has a high (low) value, so does the other. If there is a perfect linear relationship with negative slope between the two variables, we have a correlation coefficient of -1; if there is negative correlation, whenever one variable has a high (low) value; the other has a low (high) value. A correlation coefficient of 0 means that there is no linear relationship between the variables.

If we are interested only in measuring the association between the two variables, then Pearson’s Correlation Coefficient (r) gives us an estimate of the strength of the linear association between two numerical variables.

OR

Solution:

The correlation coefficient has the following properties:

For any data set, r lies between -1 and +1.

If r = +1, or -1, the linear relationship is perfect, that is, all the points lie exactly on a straight line. If r = +1, variable y increases as x increases (i.e., the line slopes upwards). If r = -1, variable y decreases as x increases (i.e., the line slopes downward).

If r = 0, there is no linear relationship between y and x. This may mean that there is no relationship at all between the two variables (i.e., knowing x tells us nothing about the value of y). (See Diagram e.) However, we could also obtain r = 0 if there were a curved relationship between y and x. (See Diagram f.)

A useful interpretation of r is that its square (r2) measures the proportion of the variability in variable b, accounted for by the linear relationship with variable x.