Question

The unit of coefficient of correlation is:

• A. Percentage
• B. Same unit of the data
• C. Ratio
• D. No unit

Hint:

Any statistic calculated as a ratio of measures with identical units (such as covariance divided by the product of standard deviations) becomes dimensionless.
This makes the correlation coefficient highly versatile for comparing different datasets.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The Pearson product-moment correlation coefficient, denoted as \(r\), is a measure of the linear association between two continuous variables, \(X\) and \(Y\).
It quantifies both the strength and direction of the linear relationship, ranging from \(-1\) to \(+1\).

Step 2: Key Formula or Approach:
The mathematical formula for the correlation coefficient \(r\) is:
\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \]
where \(\text{Cov}(X, Y)\) is the covariance between \(X\) and \(Y\), \(\sigma_X\) is the standard deviation of \(X\), and \(\sigma_Y\) is the standard deviation of \(Y\).

Step 3: Detailed Explanation:
Let us analyze the units of the numerator and the denominator of the correlation formula:
Let the variable \(X\) have a physical unit of measurement denoted by \([U_X]\) (for example, kilograms).
Let the variable \(Y\) have a physical unit of measurement denoted by \([U_Y]\) (for example, meters).
The covariance is defined as:
\[ \text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)] \]
The unit of covariance is the product of the units of \(X\) and \(Y\):
\[ \text{Unit of Cov}(X, Y) = [U_X] \times [U_Y] \]
The standard deviation \(\sigma_X\) has the same unit of measurement as the variable \(X\):
\[ \text{Unit of } \sigma_X = [U_X] \]
Similarly, the standard deviation \(\sigma_Y\) has the same unit of measurement as the variable \(Y\):
\[ \text{Unit of } \sigma_Y = [U_Y] \]
Now, we substitute these units into the expression for the correlation coefficient \(r\):
\[ \text{Unit of } r = \frac{\text{Unit of Cov}(X, Y)}{\text{Unit of } \sigma_X \times \text{Unit of } \sigma_Y} = \frac{[U_X] \times [U_Y]}{[U_X] \times [U_Y]} \]
The units in the numerator and denominator cancel out completely.
As a result, the correlation coefficient \(r\) is a pure, dimensionless number with no units.
This property allows us to compare the linear relationships of different pairs of variables, even if they are measured in entirely different units.

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