1. Range of Correlation Coefficient

• The correlation coefficient always lies between -1 and 1: $$-1 \leq r \leq 1$$−1≤r≤1
• $r = 1$r=1 → Perfect positive correlation (both variables move in the same direction).
• $r = -1$r=−1 → Perfect negative correlation (one variable increases while the other decreases).
• $r = 0$r=0 → No linear correlation (but there might be a non-linear relationship).

2. Effect of Scaling and Shifting

• Multiplying a variable by a positive constant does not change $r$. $$r(x, 2y) = r(x, y)$$
  
• Multiplying by a negative constant reverses the sign of $r$. $$r(x, -y) = -r(x, y)$$
  
• Adding or subtracting a constant to any variable does not affect $r$. $$r(x, y + c) = r(x, y)$$
  

3. Swapping Variables

• Correlation is symmetric, meaning the order does not matter. $$r(x, y) = r(y, x)$$
  

4. Correlation and Independence

• If two variables are independent, their correlation is 0.
• However, $r = 0$ does not mean independence—there could be a non-linear relationship.