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.
Similar Questions
The two regression lines y = c + mx and x = a + dy always pass through:
If the correlation coefficient is the geometric mean of regression coefficients, then:
The limits of a multiple correlation coefficient R1.23 are:
For three variables X1, X2 and X3, the following information is available:
s1 = 10, r12 = 0.90
s2 = 5, r13 = 0.75
s3 = 3, r23 = 0.70
Then partial regression coefficient (b13,2) of X2 on X1 and X3 is:If the correlation coefficient between two variables is zero, then the two lines of regression will be:
Using usual notations, the correlation coefficient between two random variables, X and Y, is given by:
In Spearman's rank correlation coefficient, squared differences of ranks are used to:
In Spearman's rank correlation coefficient:
\[ r_s = 1 - \frac{6\sum_{i=1}^n d_i^2}{n(n^2-1)} \]The maximum value of \(\sum_{i=1}^n d_i^2\) in case of untied ranks is:
The Karl Pearson’s correlation coefficient (r) between two random variables, X and Y, is computed by:
For three random variables X1 , X2 and X3, the pairwise correlation coefficients are r12 = r13 = r23 = r. The multiple correlation coefficient \(\rm R_{1.2.3}^2\) is
Important Questions from Correlation and Regression
If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
Two variates, x and y, are uncorrelated and have standard deviations σ xand σ yrespectively. What is the correlation coefficient between x + y and x – y?
If the covariance between x and y is 30, variance of x is 25 and variance of y is 144, then what is the correlation coefficient?
The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
