Correlation Coefficient Properties: Origin and Scale Invariance
The coefficient of correlation, denoted by $r$, measures the strength and direction of a linear relationship between two random variables.
A key statistical property of the correlation coefficient is its invariance under specific linear transformations of the variables.
Independence from Origin Change
If we change the origin of the variables $X$ and $Y$ by subtracting constants $a$ and $b$ respectively, creating new variables $X' = X - a$ and $Y' = Y - b$, the correlation coefficient remains unchanged. That is, $r_{X'Y'} = r_{XY}$.
Independence from Scale Change
Similarly, if we change the scale of the variables $X$ and $Y$ by multiplying by positive constants $c$ and $d$ respectively, creating new variables $X'' = cX$ and $Y'' = dY$ (where $c > 0, d > 0$), the correlation coefficient also remains unchanged. That is, $r_{X''Y''} = r_{XY}$.
Combining these, the correlation coefficient is independent of linear transformations of the form $aX+b$ and $cY+d$, provided $c$ and $d$ are non-zero.
Therefore, the coefficient of correlation is independent of both change of origin and change of scale.
Conclusion: The correct option is the one stating independence from both change of origin and scale.
Important Questions from Correlation and Regression
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
Which of the following statements is/are correct in respect of regression coefficients?
1. It measures the degree of linear relationship between two variables
2. It gives the value by which one variable changes for a unit change in the other variable.
Select the correct answer using the code given below.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 ?
If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?
