Formula
1. Regression coefficient

	Regression coefficient `y` on `x`	Regression coefficient `x` on `y`
1.	`byx = (sum (x-bar x)(y-bar y))/(sum (x-bar x)^2)`	`bxy = (sum (x-bar x)(y-bar y))/(sum (y-bar y)^2)`
2.	`byx = (n sum xy - (sum x)(sum y))/(n sum x^2 - (sum x)^2)`	`bxy = (n sum xy - (sum x)(sum y))/(n sum y^2 - (sum y)^2)`
3.	`byx = (n sum dx dy - (sum dx)(sum dy))/(n sum dx^2 - (sum dx)^2)`	`bxy = (n sum dx dy - (sum dx)(sum dy))/(n sum dy^2 - (sum dy)^2)`
4.	`byx = r * (sigma y)/(sigma x)`	`bxy = r * (sigma x)/(sigma y)`

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2. Regression Line y on x
`y - bar y = byx (x - bar x)`

3. Regression Line x on y
`x - bar x = bxy (y - bar y)`

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Example-1
1. Find Regression line equations from the following data

Class-X	Y
2 - 4	3
4 - 6	4
6 - 8	2
8 - 10	1

Solution:

Class-X	Mid value `x`	`y`	`x^2`	`y^2`	`x*y`
2 - 4	3	3	9	9	9
4 - 6	5	4	25	16	20
6 - 8	7	2	49	4	14
8 - 10	9	1	81	1	9
---	---	---	---	---	---
	`sum x=24`	`sum y=10`	`sum x^2=164`	`sum y^2=30`	`sum xy=52`

Mean `bar x = (sum x)/n`

`=24/4`

`=6`


Mean `bar y = (sum y)/n`

`=10/4`

`=2.5`


`byx = (n sum xy - (sum x)(sum y))/(n sum x^2 - (sum x)^2)`

`=(4 * 52 - 24 * 10 )/(4 * 164 - (24)^2)`

`=(208 - 240 )/(656 - 576)`

`=-32/80`

`=-0.4`


Regression Line y on x
`y - bar y = byx (x - bar x)`

`y - 2.5 = -0.4 (x - 6)`

`y - 2.5 = -0.4 x + 2.4`

`y = -0.4 x + 2.4 + 2.5`

`y = -0.4 x + 4.9`


`bxy = (n sum xy - (sum x)(sum y))/(n sum y^2 - (sum y)^2)`

`=(4 * 52 - 24 * 10 )/(4 * 30 - (10)^2)`

`=(208 - 240 )/(120 - 100)`

`=-32/20`

`=-1.6`


Regression Line x on y
`x - bar x = bxy (y - bar y)`

`x - 6 = -1.6 (y - 2.5)`

`x - 6 = -1.6 y + 4`

`x = -1.6 y + 4 + 6`

`x = -1.6 y + 10`

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