Question

If each value of variable X is multiplied by 8 and of Y is multiplied by 10, then correlation coefficient of new data is:

• A. Ten times as obtained from original data
• B. Eight times as obtained from original data
• C. Eighty times as obtained from original data
• D. Same as obtained from original data

Hint:

Exam Tip:
Correlation is scale-invariant:

• Adding or subtracting a constant does not change \(r\).
• Multiplying by a positive constant does not change \(r\).
• Multiplying by a negative constant changes the sign of \(r\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The correlation coefficient is a measure of the strength and direction of a linear relationship between two variables. It is scale-invariant.

Step 2: Key Formula or Approach:
The correlation coefficient between two variables \(X\) and \(Y\) is: \[ r = \frac{\text{Cov}(X, Y)}{\text{SD}(X) \text{SD}(Y)} \] If we transform \(X \to aX\) and \(Y \to bY\) with \(a, b > 0\), then: \[ \text{Cov}(aX, bY) = ab \, \text{Cov}(X, Y), \quad \text{SD}(aX) = a \, \text{SD}(X), \quad \text{SD}(bY) = b \, \text{SD}(Y) \] So, \[ r_{new} = \frac{ab \, \text{Cov}(X, Y)}{a \, \text{SD}(X) \cdot b \, \text{SD}(Y)} = \frac{\text{Cov}(X, Y)}{\text{SD}(X) \text{SD}(Y)} = r \]

Step 3: Detailed Explanation:
Since both variables are multiplied by positive constants, the correlation coefficient remains unchanged.
Thus, the new correlation coefficient is the same as the original.

Step 4: Final Answer:
Therefore, option (D) is correct.