Understanding Correlation Coefficient and Linear Transformations

The question asks how the correlation coefficient between two variables, X and Y, changes when both variables undergo specific linear transformations. We are given the initial correlation coefficient between X and Y, and the equations for the transformations.

The correlation coefficient, denoted by \(r_{XY}\), measures the strength and direction of the linear relationship between two variables X and Y. Its value ranges from -1 to +1.

We are given that the correlation coefficient between X and Y is \(r_{XY} = 0.6\).

The transformations are given as:

• \(U = 2 - 3X\)
• \(V = 4Y + 1\)

These are linear transformations. A linear transformation of a variable X is of the form \(U = a + bX\), where 'a' and 'b' are constants. In our case for variable X, \(a=2\) and \(b=-3\). For variable Y, the transformation is \(V = c + dY\), where \(c=1\) and \(d=4\).

Effect of Linear Transformations on Correlation Coefficient

Let's examine how these transformations affect the covariance and standard deviations, which are components of the correlation coefficient formula. The correlation coefficient between U and V is given by:

\(r_{UV} = \frac{Cov(U, V)}{\sigma_U \sigma_V}\)

Where \(Cov(U, V)\) is the covariance between U and V, \(\sigma_U\) is the standard deviation of U, and \(\sigma_V\) is the standard deviation of V.

For linear transformations \(U = a + bX\) and \(V = c + dY\):

The covariance between U and V is related to the covariance between X and Y by:

\(Cov(U, V) = Cov(a + bX, c + dY)\)

Using the properties of covariance, \(Cov(a_1 + b_1X_1, a_2 + b_2X_2) = b_1 b_2 Cov(X_1, X_2)\), we get:

\(Cov(U, V) = (b)(d) Cov(X, Y)\)

In our case, \(b = -3\) and \(d = 4\), so:

\(Cov(U, V) = (-3)(4) Cov(X, Y) = -12 Cov(X, Y)\)

The standard deviation of U is related to the standard deviation of X by:

\(\sigma_U = \sqrt{Var(U)} = \sqrt{Var(a + bX)}\)

Using the property \(Var(a + bX) = b^2 Var(X)\), we get:

\(\sigma_U = \sqrt{b^2 Var(X)} = |b| \sqrt{Var(X)} = |b| \sigma_X\)

In our case, \(b = -3\), so \(\sigma_U = |-3| \sigma_X = 3 \sigma_X\).

Similarly, the standard deviation of V is related to the standard deviation of Y by:

\(\sigma_V = \sqrt{Var(V)} = \sqrt{Var(c + dY)}\)

\(\sigma_V = \sqrt{d^2 Var(Y)} = |d| \sqrt{Var(Y)} = |d| \sigma_Y\)

In our case, \(d = 4\), so \(\sigma_V = |4| \sigma_Y = 4 \sigma_Y\).

Calculating the New Correlation Coefficient

Now substitute these into the formula for \(r_{UV}\):

\(r_{UV} = \frac{Cov(U, V)}{\sigma_U \sigma_V} = \frac{(b)(d) Cov(X, Y)}{|b| \sigma_X |d| \sigma_Y}\)

\(r_{UV} = \frac{(bd)}{|bd|} \frac{Cov(X, Y)}{\sigma_X \sigma_Y}\)

We know that \(r_{XY} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}\), so:

\(r_{UV} = \frac{(bd)}{|bd|} r_{XY}\)

The term \(\frac{(bd)}{|bd|}\) is equal to +1 if b and d have the same sign, and -1 if b and d have opposite signs. In this problem, \(b = -3\) (negative) and \(d = 4\) (positive). Since they have opposite signs, \(\frac{(bd)}{|bd|} = -1\).

Therefore, \(r_{UV} = -r_{XY}\).

Given \(r_{XY} = 0.6\), the correlation coefficient between U and V is:

\(r_{UV} = -(0.6) = -0.6\)

The correlation coefficient between the transformed variables U and V is -0.6.

This result shows that while scaling and shifting variables linearly does not change the magnitude of the correlation coefficient (it remains 0.6), it can change its sign if the scaling factors (the 'b' and 'd' values) have opposite signs.

Revision Table: Correlation Transformation Summary

Additional Information: Properties of Correlation

The correlation coefficient is a powerful statistical measure with several important properties:

• Range: The value of the correlation coefficient \(r\) is always between -1 and +1, inclusive (\(-1 \le r \le +1\)).
• Direction: A positive correlation (\(r > 0\)) indicates that as one variable increases, the other variable tends to increase. A negative correlation (\(r < 0\)) indicates that as one variable increases, the other variable tends to decrease.
• Strength: The closer the absolute value of \(r\) is to 1, the stronger the linear relationship. A value of 0 indicates no linear relationship.
• Invariance to Scale and Location (with same sign): Adding a constant to a variable (location shift) or multiplying a variable by a positive constant (scale) does not change the correlation coefficient with another variable. That is, if \(U = a + bX\) with \(b > 0\), then \(r_{UY} = r_{XY}\). Similarly, if \(V = c + dY\) with \(d > 0\), then \(r_{XV} = r_{XY}\).
• Sign Change with Negative Scaling: If a variable is multiplied by a negative constant, the sign of the correlation coefficient with another variable is reversed. If \(U = a + bX\) with \(b < 0\), then \(r_{UY} = -r_{XY}\).
• Units: The correlation coefficient is a dimensionless quantity, meaning it does not have units. It is just a number.

Understanding these properties, especially the effect of linear transformations, is crucial when working with statistical data.