If the correlation between X and Y is 0.3, then correlation coefficient between 2X and 3Y is:
0.3
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
| Property | Rule/Formula | Explanation |
|---|---|---|
| Range | \( -1 \le \rho \le 1 \) | Indicates the strength and direction of the linear relationship. |
| Effect of Linear Transformation | \( \rho(aX+b, cY+d) = \frac{ac}{|ac|} \rho(X, Y) \) | Magnitude is unchanged. Sign changes only if \( a \) and \( c \) have different signs. Constants \( b \) and \( d \) have no effect on correlation. |
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.
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