Understanding Correlation Coefficient and Linear Transformations

The correlation coefficient, often denoted by \( \rho \) (rho) for a population or \( r \) for a sample, is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to +1. A value of 0.3 indicates a weak positive linear relationship between variables X and Y.

The question asks how this correlation changes if we transform the variables linearly, specifically by multiplying X by 2 and Y by 3.

Impact of Linear Transformations on Correlation

A key property of the correlation coefficient is its behavior under linear transformations. If we transform a variable \( X \) into \( aX + b \) and a variable \( Y \) into \( cY + d \), the correlation between the transformed variables \( aX+b \) and \( cY+d \) is related to the original correlation between \( X \) and \( Y \) by the following formula:

$$ \rho(aX+b, cY+d) = \frac{ac}{|ac|} \rho(X, Y) $$

This formula shows that the magnitude of the correlation coefficient remains the same, regardless of the scaling factors \( a \) and \( c \) or the constants \( b \) and \( d \). The only thing that can change is the sign, which happens if exactly one of the scaling factors \( a \) or \( c \) is negative (i.e., \( ac < 0 \)). If both \( a \) and \( c \) have the same sign (i.e., \( ac > 0 \)), the term \( \frac{ac}{|ac|} \) is +1, and the correlation sign remains unchanged. If \( ac < 0 \), the term \( \frac{ac}{|ac|} \) is -1, and the correlation sign is reversed.

Step-by-Step Calculation

Given the original correlation between X and Y is \( \rho(X, Y) = 0.3 \).

We are asked to find the correlation between 2X and 3Y.

Comparing the transformations \( 2X \) with \( aX + b \) and \( 3Y \) with \( cY + d \), we can identify the values:

• For the transformation of X: \( a = 2 \) and \( b = 0 \).
• For the transformation of Y: \( c = 3 \) and \( d = 0 \).

Now, we apply the formula for the correlation between linearly transformed variables:

$$ \rho(2X, 3Y) = \frac{(2)(3)}{|(2)(3)|} \rho(X, Y) $$

First, calculate the product \( ac \): \( (2)(3) = 6 \).

The absolute value of \( ac \) is \( |6| = 6 \).

Substitute these values and the given original correlation (\( \rho(X, Y) = 0.3 \)) into the formula:

$$ \rho(2X, 3Y) = \frac{6}{6} \times 0.3 $$

$$ \rho(2X, 3Y) = 1 \times 0.3 $$

$$ \rho(2X, 3Y) = 0.3 $$

Since the scaling factors (2 and 3) are both positive, their product is positive, and the term \( \frac{ac}{|ac|} \) is +1. Therefore, the correlation coefficient retains its original sign and magnitude.

Correlation Coefficient Revision Table

Additional Information on Statistical Measures

While the correlation coefficient is invariant (except possibly for sign) under linear transformations, other statistical measures behave differently.

• Mean: The mean of \( aX+b \) is \( a \) times the mean of \( X \) plus \( b \): \( E(aX+b) = a E(X) + b \).
• Variance: The variance of \( aX+b \) is \( a^2 \) times the variance of \( X \): \( Var(aX+b) = a^2 Var(X) \). The constant \( b \) does not affect variance.
• Standard Deviation: The standard deviation of \( aX+b \) is the absolute value of \( a \) times the standard deviation of \( X \): \( SD(aX+b) = |a| SD(X) \). The constant \( b \) does not affect standard deviation.

Understanding how different statistical measures respond to linear transformations is crucial in data analysis.