The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
0.6
Understanding Correlation and Regression
In statistics, the relationship between two variables is often studied using correlation and regression analysis. The coefficient of correlation (\(r\)) measures the strength and direction of a linear relationship between two variables, while the coefficients of regression (\(b_{xy}\) and \(b_{yx}\)) describe the average change in one variable per unit change in the other.
Relationship Between Correlation and Regression Coefficients
There is a specific mathematical relationship connecting the coefficient of correlation and the two coefficients of regression. If we denote the regression coefficient of Y on X as \(b_{yx}\) and the regression coefficient of X on Y as \(b_{xy}\), the coefficient of correlation \(r\) is the geometric mean of the two regression coefficients. The formula is given by:
\(r = \pm \sqrt{b_{xy} \times b_{yx}}\)
It is important to note that the sign of the coefficient of correlation (\(r\)) must be the same as the sign of both regression coefficients (\(b_{xy}\) and \(b_{yx}\)). If \(b_{xy}\) and \(b_{yx}\) are both positive, \(r\) must be positive. If \(b_{xy}\) and \(b_{yx}\) are both negative, \(r\) must be negative. If they have different signs, it indicates an error, as regression coefficients derived from the same data set must have the same sign.
Calculating the Coefficient of Correlation
Given the coefficients of regression are 0.2 and 1.8, we can assign them to \(b_{xy}\) and \(b_{yx}\). Let's assume \(b_{xy} = 0.2\) and \(b_{yx} = 1.8\). Since both coefficients are positive, the coefficient of correlation \(r\) will also be positive.
Using the formula:
\(r = \sqrt{b_{xy} \times b_{yx}}\)
Substitute the given values:
\(r = \sqrt{0.2 \times 1.8}\)
First, calculate the product of the regression coefficients:
\(0.2 \times 1.8 = 0.36\)
Now, find the square root of the product:
\(r = \sqrt{0.36}\)
\(r = 0.6\)
The calculated coefficient of correlation is 0.6.
We can verify if this value is valid. The coefficient of correlation \(r\) must lie between -1 and +1 (inclusive). Our calculated value 0.6 falls within this range, confirming its validity.
Summary of Calculation Steps
- Identify the given regression coefficients: \(b_{xy} = 0.2\) and \(b_{yx} = 1.8\).
- Determine the sign of the correlation coefficient. Since both regression coefficients are positive, \(r\) is positive.
- Apply the formula \(r = \sqrt{b_{xy} \times b_{yx}}\).
- Calculate the product of the regression coefficients: \(0.2 \times 1.8 = 0.36\).
- Find the square root of the product: \(\sqrt{0.36} = 0.6\).
- The coefficient of correlation is 0.6.
| Statistic | Symbol | Description | Relationship to Others |
|---|---|---|---|
| Coefficient of Correlation | \(r\) | Measures strength and direction of linear relationship (-1 to +1) | \(r = \pm \sqrt{b_{xy} \times b_{yx}}\) |
| Regression Coefficient (Y on X) | \(b_{yx}\) | Change in Y for a unit change in X | \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\) |
| Regression Coefficient (X on Y) | \(b_{xy}\) | Change in X for a unit change in Y | \(b_{xy} = r \frac{\sigma_x}{\sigma_y}\) |
Revision Table: Key Formulas
| Concept | Formula |
|---|---|
| Relationship between \(r\), \(b_{xy}\), \(b_{yx}\) | \(r = \pm \sqrt{b_{xy} \times b_{yx}}\) |
| Regression Coefficient \(b_{yx}\) | \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\) or \(b_{yx} = \frac{\text{Cov}(X,Y)}{\sigma_x^2}\) |
| Regression Coefficient \(b_{xy}\) | \(b_{xy} = r \frac{\sigma_x}{\sigma_y}\) or \(b_{xy} = \frac{\text{Cov}(X,Y)}{\sigma_y^2}\) |
Additional Information: Properties
Here are some additional points about correlation and regression coefficients:
- The sign of the coefficient of correlation \(r\) is always the same as the sign of both regression coefficients \(b_{xy}\) and \(b_{yx}\).
- If \(r=0\), the variables are uncorrelated, and the regression lines are perpendicular to each other (one horizontal, one vertical, assuming axes are scaled equally).
- If \(r=1\) or \(r=-1\), there is perfect correlation, and the two regression lines are identical and coincide.
- The geometric mean of the two regression coefficients \(b_{xy}\) and \(b_{yx}\) cannot be greater than their arithmetic mean. Also, the product \(b_{xy} \times b_{yx}\) must be less than or equal to 1 (or \(|r| \le 1\)). In this case, \(0.2 \times 1.8 = 0.36\), and \(\sqrt{0.36}=0.6\), which respects the property \(r^2 \le 1\).
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?
Consider the following statements:
1. The regression line of y on x is \(\rm y = \frac{3}{4}x+2\)
2. The regression line of x on y is \(\rm x = \frac{3}{4}y+\frac{1}{4}\)
Which of the above statements is/are correct?
