Question

The range of Correlation coefficient is
1. -1 to +1
2. -1 to 0
3. 0 to +1
4. -\(\infty\) to +\(\infty\)

• A. -1 to +1
• B. -1 to 0
• C. 0 to +1
• D. -\(\infty\) to +\(\infty\)

Hint:

Because correlation is a standardized measure of association, its value is strictly bounded between \( -1 \) and \( +1 \). Any calculated value outside this interval indicates an arithmetic error.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the mathematical boundaries and range of Pearson's product-moment correlation coefficient (\( r \)).

Step 2: Detailed Explanation:
Pearson's correlation coefficient (\( r \)) measures the strength and direction of the linear relationship between two continuous variables \( X \) and \( Y \). It is defined as:
\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \] According to the Cauchy-Schwarz Inequality, for any two real-valued random variables with finite variances, the absolute value of their covariance is bounded by the product of their standard deviations:
\[ |\text{Cov}(X, Y)| \le \sigma_X \sigma_Y \] Dividing both sides by \( \sigma_X \sigma_Y \):
\[ \frac{|\text{Cov}(X, Y)|}{\sigma_X \sigma_Y} \le 1 \implies |r| \le 1 \] This inequality establishes the range of the correlation coefficient:
\[ -1 \le r \le +1 \] The interpretation of the values within this range is as follows:
- \( r = +1 \): Perfect positive linear relationship.
- \( r = -1 \): Perfect negative linear relationship.
- \( r = 0 \): No linear relationship between the variables.

Step 3: Final Answer:
The range of the correlation coefficient is from -1 to +1.
Therefore, the correct choice is Option (A).