The correlation coefficient between two variables X and Y is 0.4. The correlation coefficient between 2X and (-Y) will be:
-0.4
Understanding Correlation and Linear Transformations
The correlation coefficient, often denoted by \( r \), is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables, say X and Y. Its value ranges from -1 to +1. A positive value indicates a positive linear relationship (as one variable increases, the other tends to increase), a negative value indicates a negative linear relationship (as one variable increases, the other tends to decrease), and a value of 0 indicates no linear relationship.
Effect of Linear Transformations on Correlation
Linear transformations involve changing the variables by adding a constant, multiplying by a constant, or both. A linear transformation of a variable X can be written as \( X' = aX + b \), where \( a \) and \( b \) are constants. Similarly, for variable Y, it can be written as \( Y' = cY + d \).
The correlation coefficient between two variables is not affected by changes in the origin (adding constants \( b \) and \( d \)). However, it is affected by changes in the scale and, importantly, by changes in the sign of the variables.
Formula for Correlation under Linear Transformation
If the correlation coefficient between X and Y is \( r_{XY} \), and we transform the variables to \( X' = aX + b \) and \( Y' = cY + d \), the correlation coefficient between the new variables \( X' \) and \( Y' \) is given by the formula:
\( r_{X'Y'} = \frac{ac}{|a||c|} r_{XY} \)
This formula shows that the sign of the new correlation coefficient \( r_{X'Y'} \) depends on the signs of the constants \( a \) and \( c \). If \( a \) and \( c \) have the same sign (both positive or both negative), then \( ac \) and \( |a||c| \) will both be positive, and the term \( \frac{ac}{|a||c|} \) will be +1, meaning \( r_{X'Y'} = r_{XY} \). If \( a \) and \( c \) have opposite signs (one positive and one negative), then \( ac \) will be negative and \( |a||c| \) will be positive, making the term \( \frac{ac}{|a||c|} \) equal to -1, and thus \( r_{X'Y'} = -r_{XY} \).
Calculating the New Correlation Coefficient
In the given problem, the original correlation coefficient between X and Y is \( r_{XY} = 0.4 \). We are asked to find the correlation coefficient between the variables 2X and (-Y).
Let the transformed variables be \( X' = 2X \) and \( Y' = -Y \).
Comparing these to the general linear transformation form \( X' = aX + b \) and \( Y' = cY + d \), we can identify the constants:
- For \( X' = 2X \), we have \( a = 2 \) and \( b = 0 \).
- For \( Y' = -Y \), we have \( c = -1 \) and \( d = 0 \).
Now, we apply the formula for the correlation coefficient of transformed variables:
\( r_{X'Y'} = \frac{ac}{|a||c|} r_{XY} \)
Substitute the values of \( a \), \( c \), and \( r_{XY} \):
\( r_{(2X)(-Y)} = \frac{(2)(-1)}{|2||-1|} (0.4) \)
\( r_{(2X)(-Y)} = \frac{-2}{(2)(1)} (0.4) \)
\( r_{(2X)(-Y)} = \frac{-2}{2} (0.4) \)
\( r_{(2X)(-Y)} = -1 \times (0.4) \)
\( r_{(2X)(-Y)} = -0.4 \)
Therefore, the correlation coefficient between 2X and (-Y) is -0.4.
This result makes intuitive sense: multiplying X by a positive constant (2) scales the X values but does not change the direction of the linear relationship with Y. However, multiplying Y by a negative constant (-1) reverses the direction of the relationship with X. If X and Y were positively correlated (tend to increase or decrease together), then 2X and -Y will be negatively correlated (as 2X increases, -Y tends to decrease, meaning Y tends to increase). The strength of the relationship remains the same (0.4), only the direction changes (from positive to negative).
Revision Table: Key Concepts
| Concept | Description | Effect on Correlation |
|---|---|---|
| Correlation Coefficient (\( r \)) | Measures strength and direction of linear relationship (-1 to +1). | Base value. |
| Change of Origin (Add/Subtract Constant) | \( X' = X + b \) | No change (\( r_{X'Y} = r_{XY} \)). |
| Change of Scale (Multiply by Positive Constant) | \( X' = aX \) (where \( a > 0 \)) | No change (\( r_{X'Y} = r_{XY} \)). |
| Change of Sign (Multiply by Negative Constant) | \( X' = aX \) (where \( a < 0 \)) | Sign reverses (\( r_{X'Y} = -r_{XY} \)). |
| Combined Linear Transformation | \( X' = aX + b \), \( Y' = cY + d \) | \( r_{X'Y'} = \frac{ac}{|a||c|} r_{XY} \) |
Additional Information: Properties of Correlation Coefficient
Beyond linear transformations, the correlation coefficient has several important properties:
- The correlation coefficient is a dimensionless quantity; it does not have units.
- Correlation only measures the strength of a linear relationship. Variables can be strongly non-linearly related even if their correlation coefficient is close to 0.
- Correlation does not imply causation. A high correlation between two variables does not mean that one variable causes the other. There might be a confounding variable or the relationship could be coincidental.
- The correlation coefficient is symmetric: \( r_{XY} = r_{YX} \).
- The correlation coefficient is sensitive to outliers, which can significantly affect its value.
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