Question

Value of correlation coefficient is from:

• A. + 1 through 0 to – 1
• B. 1 to 0
• C. + infinity through 0 to – infinity
• D. A to Z

Hint:

The correlation coefficient (\( r \)) ranges from \( -1 \) to \( +1 \).
The coefficient of determination (\( R^2 \)), which is the square of the correlation coefficient, ranges from \( 0 \) to \( 1 \).

Answer 1

The Correct Option is A

In statistics, the correlation coefficient (specifically Pearson's product-moment correlation coefficient, denoted as \( r \)) measures the strength and direction of the linear relationship between two continuous variables.
It is a dimensionless index that provides a standardized measure of how two variables change together.

Step 2: Detailed Explanation:
The mathematical definition of the correlation coefficient ensures that its value is bounded between positive one and negative one:
\[ -1 \le r \le +1 \]
The interpretation of the values within this range is as follows:
- Value of \( +1 \): Indicates a perfect positive linear relationship.
As one variable increases, the other variable increases in a perfectly proportional manner.
- Value of \( 0 \): Indicates the absence of any linear relationship between the two variables.
Changes in one variable do not predict any linear change in the other.
- Value of \( -1 \): Indicates a perfect negative linear relationship.
As one variable increases, the other variable decreases in a perfectly proportional manner.
Because the value is normalized by dividing the covariance of the two variables by the product of their standard deviations, it cannot exceed the absolute value of $1$.
Therefore, the range of possible values for a correlation coefficient is strictly from \( +1 \) through \( 0 \) to \( -1 \).

Step 3: Final Answer:
The correct range of values is given by Option (A).