Question

Correlation coefficient is not affected by
• [(i)] Interchanging the two variables
• [(ii)] Adding the same number to all the values of one variable
• [(iii)] Multiplying all the values of one variable by the same positive number
• [(iv)] Multiplying all the values of one variable by the same negative number Choose the correct answer from the following:

• A. All the four statements are correct
• B. Statements (i) and (ii) are correct
• C. Statements (i), (ii) and (iii) are correct
• D. Statements (ii), (iii) and (iv) are correct

Hint:

Correlation is unaffected by change of origin and positive change of scale, but changes sign under multiplication by a negative constant.

Answer 1

The Correct Option is C

Concept: Pearson's correlation coefficient \(r\) measures the strength and direction of linear association between two variables. \[ r=\frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \] Important properties:
• Interchanging \(X\) and \(Y\) does not change \(r\).
• Adding a constant does not change \(r\).
• Multiplying by a positive constant does not change \(r\).
• Multiplying by a negative constant changes only the sign of \(r\).

Step 1: Examine Statement (i).
\[ r(X,Y)=r(Y,X) \] Hence Statement (i) is correct.

Step 2: Examine Statement (ii).
If \[ X'=X+9 \] then correlation remains unchanged. Hence Statement (ii) is correct.

Step 3: Examine Statement (iii).
If \[ X'=kX,\quad k>0 \] correlation coefficient remains unchanged. Hence Statement (iii) is correct.

Step 4: Examine Statement (iv).
If \[ X'=-X \] then \[ r(X',Y)=-r(X,Y) \] The sign changes. Therefore Statement (iv) is not correct.

Step 5: Choose the answer.
Correct statements are: \[ (i),\ (ii),\ (iii) \] Hence, \[ \boxed{\text{Option (C)}} \]