Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : If X and Y are any two variables and are transformed to the new variables U and V defined by $U = \frac{X - A}{h}$, $V = \frac{Y - B}{K}$ where A, B, h, K are any constants, h and K > 0 then, $r_{xy} = r_{uv}$ Reason (R) :Correlation coefficient(r) is independent of change of origin and scale. In the light of the above statements, choose the most appropriate answer from the options given below :
Correlation Coefficient: Independence of Transformations
This question examines the properties of the correlation coefficient ($r$) concerning linear transformations of variables.
Evaluating Assertion (A)
Assertion (A) states that for variables X and Y, transformed into U and V by $U = \frac{X - A}{h}$ and $V = \frac{Y - B}{K}$ (where $h, K > 0$), the correlation coefficient remains the same: $r_{xy} = r_{uv}$.
To verify this, we use the formula for the correlation coefficient: $r = \frac{\text{Cov}(X, Y)}{\sigma_x \sigma_y}$.
The covariance of the transformed variables is:
$\text{Cov}(U, V) = \text{Cov}\left(\frac{X - A}{h}, \frac{Y - B}{K}\right) = \frac{1}{hK} \text{Cov}(X, Y)$
The standard deviations of the transformed variables are:
$\sigma_U = \frac{1}{h} \sigma_x$ (since $h > 0$)
$\sigma_V = \frac{1}{K} \sigma_y$ (since $K > 0$)
The correlation coefficient for U and V is:
$r_{uv} = \frac{\text{Cov}(U, V)}{\sigma_U \sigma_V} = \frac{\frac{1}{hK} \text{Cov}(X, Y)}{\left(\frac{1}{h} \sigma_x\right) \left(\frac{1}{K} \sigma_y\right)} = \frac{\frac{1}{hK} \text{Cov}(X, Y)}{\frac{1}{hK} \sigma_x \sigma_y} = \frac{\text{Cov}(X, Y)}{\sigma_x \sigma_y} = r_{xy}$
Thus, Assertion (A) is correct.
Evaluating Reason (R)
Reason (R) states that the correlation coefficient ($r$) is independent of change of origin and scale. This is a fundamental property of the correlation coefficient. A change in origin shifts the data without altering the spread or relationship, and a change in scale stretches or compresses the data but preserves the relative linear association. The transformation in Assertion (A) involves both a change of origin (A, B) and a change of scale (h, K).
Therefore, Reason (R) is correct.
Assessing the Explanation
Assertion (A) demonstrates a specific case where the correlation coefficient remains unchanged under linear transformations involving changes of origin and scale. Reason (R) provides the general principle that explains why this happens. The independence from origin and scale (Reason R) is the direct reason why $r_{xy} = r_{uv}$ (Assertion A).
Hence, Reason (R) is the correct explanation of Assertion (A).
Important Questions from Correlation and Regression
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
Which of the following statements is/are correct in respect of regression coefficients?
1. It measures the degree of linear relationship between two variables
2. It gives the value by which one variable changes for a unit change in the other variable.
Select the correct answer using the code given below.If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?
