Prove that the correlation coefficient r lies between -1 and +1.
To prove that the correlation coefficient (r) lies between -1 and +1, we need to establish certain properties of r.
Property 1: r represents the strength and direction of the linear relationship between two variables.
Property 2: r can only take values from -1 to +1, inclusive.
Let's go through the proof step by step:
Step 1: Understanding the correlation coefficient
The correlation coefficient (r) quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to +1 and indicates the degree to which the variables are associated. A positive value of r indicates a positive linear relationship, a negative value indicates a negative linear relationship, and a value of 0 indicates no linear relationship.
Step 2: Proving that -1 ≤ r ≤ +1
To prove that the correlation coefficient lies between -1 and +1, we need to show that r cannot exceed these bounds.
Case 1: r ≤ +1
To prove that r ≤ +1, we assume the maximum possible positive value for r and demonstrate that it cannot exceed +1. Let's assume r = +1.
When r = +1, it means there is a perfect positive linear relationship between the variables. In this case, as the values of one variable increase, the values of the other variable increase proportionally. This represents the highest positive correlation achievable.
As per Property 2, r can take a maximum value of +1, so r ≤ +1 holds.
Case 2: r ≥ -1
To prove that r ≥ -1, we assume the maximum possible negative value for r and demonstrate that it cannot fall below -1. Let's assume r = -1.
When r = -1, it means there is a perfect negative linear relationship between the variables. In this case, as the values of one variable increase, the values of the other variable decrease proportionally. This represents the highest negative correlation achievable.
As per Property 2, r can take a minimum value of -1, so r ≥ -1 holds.
Step 3: Concluding the proof
By showing that r ≤ +1 (Case 1) and r ≥ -1 (Case 2), we have established that the correlation coefficient r lies between -1 and +1.
Therefore, we have proven that the correlation coefficient r lies between -1 and +1.