(i) The line of regression of `y` on `x` or regression line of `y` on `x` is given by
`y- bar y = r (sigma_y)/(sigma_x) (X- bar x)`
(ii) The line of regression of `x` on `y` or regression line of `x` on `y` is given by `x-bar x= r (sigma_x)/(sigma_y) (y- bar y)`
Here r is correlation coefficient
`r(x,y) =(cov (x,y))/(sigma_x sigma_y)`
(iii) Regression coefficient of `y` on `x`, is denoted by `yx`,
`b_(yx)= r (sigma_y)/(sigma_x) =(cov (x,y))/(sigma_x^2)`
(iv) Regression coefficient of `x` on `y`, is denoted by `xy`,
`b_(xy) = r (sigma_x)/(sigma_y) =(cov (x,y))/(sigma_y^2)`
(v) If `theta` is the angle between the two regression lines, then
`tan theta =((1-r^2))/(|r|) * (sigma_x sigma_y)/(sigma_x^2 + sigma_y^2)`
where , `tan theta=(M_2-M_1)/(1+ M_1 M_2)`
(a) If `r = 0, theta = pi/2`, then the two regression lines are perpendicular to each other.
(b) If `r =1` or `- 1, theta = 0, pi` then the regression lines coincide.
Coefficient of Correlation :
1. Karl Pearson's Coefficient of Correlation :
Covariance `(x, y) cov (x, y)`
`=1/n sum_(i=1)^n (x_i - bar x) (y_i - bar y) = 1/n sum_(i=1)^n x_i y_i - bar x bar y`
Let `sigma_x` and `sigma_r` the the SD of variables `x` and `y`, respectively. Then coefficient of correlation
`r(x,y) =(cov (x,y))/(sigma_x sigma_y) = (sum_(i=1)^n (x_i - bar x)(y_i - bar y))/(sqrt(sum_(i=1)^n (x_i - bar x)^2 sum_(i=1)^n (y_i - bar y)^2))`
`= (n sum x_i y_i -(sum x_i)(sum y_i))/(sqrt((n sum x_i^2 -(sum x_i)^2)(n sum y_i^2 -(sum y_i)^2))`
In general
` color{red} {b_(yx) = ( sum xy - 1/n sum x sum y)/( sum x^2 - 1/n ( sum x)^2)}`
2. Rank Correlation (Spearman's) Let `d` be the difference between paired ranks and `n` be the number of items ranked. Then, `rho` the coefficient of rank correlation is given by `rho = 1- (6 sum_(i=1)^n d^2)/(n(n^2-1))`
(i) The line of regression of `y` on `x` or regression line of `y` on `x` is given by
`y- bar y = r (sigma_y)/(sigma_x) (X- bar x)`
(ii) The line of regression of `x` on `y` or regression line of `x` on `y` is given by `x-bar x= r (sigma_x)/(sigma_y) (y- bar y)`
Here r is correlation coefficient
`r(x,y) =(cov (x,y))/(sigma_x sigma_y)`
(iii) Regression coefficient of `y` on `x`, is denoted by `yx`,
`b_(yx)= r (sigma_y)/(sigma_x) =(cov (x,y))/(sigma_x^2)`
(iv) Regression coefficient of `x` on `y`, is denoted by `xy`,
`b_(xy) = r (sigma_x)/(sigma_y) =(cov (x,y))/(sigma_y^2)`
(v) If `theta` is the angle between the two regression lines, then
`tan theta =((1-r^2))/(|r|) * (sigma_x sigma_y)/(sigma_x^2 + sigma_y^2)`
where , `tan theta=(M_2-M_1)/(1+ M_1 M_2)`
(a) If `r = 0, theta = pi/2`, then the two regression lines are perpendicular to each other.
(b) If `r =1` or `- 1, theta = 0, pi` then the regression lines coincide.
Coefficient of Correlation :
1. Karl Pearson's Coefficient of Correlation :
Covariance `(x, y) cov (x, y)`
`=1/n sum_(i=1)^n (x_i - bar x) (y_i - bar y) = 1/n sum_(i=1)^n x_i y_i - bar x bar y`
Let `sigma_x` and `sigma_r` the the SD of variables `x` and `y`, respectively. Then coefficient of correlation
`r(x,y) =(cov (x,y))/(sigma_x sigma_y) = (sum_(i=1)^n (x_i - bar x)(y_i - bar y))/(sqrt(sum_(i=1)^n (x_i - bar x)^2 sum_(i=1)^n (y_i - bar y)^2))`
`= (n sum x_i y_i -(sum x_i)(sum y_i))/(sqrt((n sum x_i^2 -(sum x_i)^2)(n sum y_i^2 -(sum y_i)^2))`
In general
` color{red} {b_(yx) = ( sum xy - 1/n sum x sum y)/( sum x^2 - 1/n ( sum x)^2)}`
2. Rank Correlation (Spearman's) Let `d` be the difference between paired ranks and `n` be the number of items ranked. Then, `rho` the coefficient of rank correlation is given by `rho = 1- (6 sum_(i=1)^n d^2)/(n(n^2-1))`