1. What is the Correlation Coefficient?

The correlation coefficient (r) is a statistical measure that quantifies the strength and direction of the linear relationship between two variables, X and Y.

2. Formula for Correlation Coefficient ($r$):

3. Properties of Coefficient of Correlation (r):

1. Range of Values:

   • The value of r lies between -1 and +1

   • −1 ≤ r ≤ +1

2. Direction of Relationship:

   • r > 0 → Positive correlation (both variables increase or decrease together)

   • r < 0 → Negative correlation (one variable increases, the other decreases)

   • r = 0 → No correlation (no linear relationship)

3. Perfect Correlation:

   • r = +1 → Perfect positive linear relationship

   • r = –1 → Perfect negative linear relationship

4. Unit-Free Measure:

   • Correlation coefficient is dimensionless; it has no units.

   • It measures the strength and direction of a relationship, not its scale.

5. Symmetry:

   • The correlation between X and Y is the same as between Y and X

   • rXY​=rYX​

6. Unaffected by Change of Origin and Scale (for linear transformations):

   • Changing the origin (adding/subtracting a constant) or scale (multiplying/dividing by a constant) of variables does not affect r.

   • Example: r will remain the same if X is transformed to 2X + 5.

7. Linear Relationship Only:

   • Correlation coefficient only measures linear relationships.

   • Non-linear relationships may exist even if r is close to 0.

8. Not Proof of Causation:

   • A high correlation does not imply causation.

   • Example: Ice cream sales and drowning may be correlated (due to summer) but one does not cause the other.

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