Linear Regression
Formula
Regression coefficients
The regression coefficient of 𝒚 on 𝒙 is denoted by 𝒃𝒚𝒙 .
𝑏𝑦𝑥 =
𝑏𝑦𝑥 =
𝑏𝑦𝑥 =
1
𝑛
1
∑ 𝑥 2 − (∑ 𝑥)2
𝑛
∑ 𝑥𝑦− ∑ 𝑥 ∑ 𝑦
∑(𝑥−𝑥̅ )(𝑦−𝑦̅)
∑(𝑥−𝑥̅ )2
1
𝑛
1
∑ 𝑢2 − (∑ 𝑢)2
𝑛
∑ 𝑢𝑣− ∑ 𝑢 ∑ 𝑣
(We use when 𝑥 , 𝑦 are small numbers)
(When 𝑥 − 𝑥̅ , 𝑦 − 𝑦̅ are small fraction less numbers)
(when 𝑢 = 𝑥 − 𝐴 , 𝑣 = 𝑦 − 𝐵 , A and B are assumed
means)
𝑏𝑦𝑥 = 𝑟.
𝜎𝑦
𝜎𝑥
(Where 𝜎𝑥 is the standard deviation of 𝑥-variate, 𝜎𝑦 is
the standard deviation of 𝑦-variate and 𝑟 is the coefficient of
correlation)
The regression coefficient of 𝒙 on 𝒚 is denoted by 𝒃𝒙𝒚 .
𝑏𝑥𝑦 =
𝑏𝑥𝑦 =
𝑏𝑥𝑦 =
1
𝑛
1
∑ 𝑦 2 − (∑ 𝑦)2
𝑛
∑ 𝑥𝑦− ∑ 𝑥 ∑ 𝑦
∑(𝑥−𝑥̅ )(𝑦−𝑦̅)
∑(𝑦−𝑦̅)2
1
𝑛
1
∑ 𝑣 2 − (∑ 𝑣)2
𝑛
∑ 𝑢𝑣− ∑ 𝑢 ∑ 𝑣
(We use when 𝑥 , 𝑦 are small numbers)
(When 𝑥 − 𝑥̅ , 𝑦 − 𝑦̅ are small fraction less numbers)
(when 𝑢 = 𝑥 − 𝐴 , 𝑣 = 𝑦 − 𝐵 , A and B are assumed
means)
𝑏𝑥𝑦 = 𝑟.
𝜎𝑥
𝜎𝑦
(Where 𝜎𝑥 is the standard deviation of 𝑥-variate, 𝜎𝑦 is
the standard deviation of 𝑦-variate and 𝑟 is the coefficient of
correlation)
Tapati's Classes
Co-efficient of correlation 𝒓(𝒙, 𝒚) 𝒐𝒓 𝝆(𝒙, 𝒚)
𝑟 2 = 𝑏𝑦𝑥 . 𝑏𝑥𝑦
and
0 ≤ 𝑟2 ≤ 1
𝑟 = √𝑏𝑦𝑥 . 𝑏𝑥𝑦
and
−1 ≤ 𝑟 ≤ 1
𝑏𝑥𝑦 , 𝑏𝑦𝑥 𝑎𝑛𝑑 𝜌(𝑥, 𝑦) are of same sign.
Equations of two lines of regression
The regression equation of 𝑦 on 𝑥 is
𝑦 − 𝑦̅ = 𝑏𝑦𝑥 (𝑥 − 𝑥̅ )
The regression equation of 𝑥 on 𝑦 is
𝑥 − 𝑥̅ = 𝑏𝑥𝑦 (𝑦 − 𝑦̅)
The two regression line intersect at (𝑥̅ , 𝑦̅)
The acute angle 𝜽 between two regression lines is given by
tan 𝜃 = |
1− 𝑟 2
𝑏𝑥𝑦 + 𝑏𝑦𝑥
|
If two lines coincide then 𝜃 = 0 .
So 1 − 𝑟 2 = 0
and 𝑟 = ±1
Tapati's Classes