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

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.