Formula

1. Regression coefficient `y` on `x` `byx = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdx^2 - (sum fdx)^2) * (h_y)/(h_x)`

2. Regression coefficient `x` on `y` `bxy = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdy^2 - (sum fdy)^2) * (h_x)/(h_y)`

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

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

---

Examples
1. Find Regression line equations from the following data

Class-Y Class-X	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60
15 - 25	6	3	0	0	0
25 - 35	3	16	10	0	0
35 - 45	0	10	15	7	0
45 - 55	0	0	7	10	4
55 - 65	0	0	0	4	5

Solution:

		C.I.`(y)`	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60
		M.V.`(y)`	15 `15=(10+20)/2`	25 `25=(20+30)/2`	35 `35=(30+40)/2`	45 `45=(40+50)/2`	55 `55=(50+60)/2`
C.I.`(x)`	M.V.`(x)`	`dy` `dx`	-2 `-2=(15-35)/10` `dy=(y-35)/10`	-1 `-1=(25-35)/10` `dy=(y-35)/10`	0 `0=(35-35)/10` `dy=(y-35)/10`	1 `1=(45-35)/10` `dy=(y-35)/10`	2 `2=(55-35)/10` `dy=(y-35)/10`	`f_x`	`fdx`	`fdx^2`	`fdxdy`
15 - 25	20 `20=(15+25)/2`	-2 `-2=(20-40)/10` `dx=(x-40)/10`	[24] `24=6*-2*-2` `f*dx*dy`6	[6] `6=3*-2*-1` `f*dx*dy`3	[0] `0=0*-2*0` `f*dx*dy`0	[0] `0=0*-2*1` `f*dx*dy`0	[0] `0=0*-2*2` `f*dx*dy`0	9 `9=6+3+0+0+0`	-18 `-18=9*-2` `fdx=f_x*dx`	36 `36=-18*-2` `fdx^2=fdx*dx`	30 `30=24+6+0+0+0`
25 - 35	30 `30=(25+35)/2`	-1 `-1=(30-40)/10` `dx=(x-40)/10`	[6] `6=3*-1*-2` `f*dx*dy`3	[16] `16=16*-1*-1` `f*dx*dy`16	[0] `0=10*-1*0` `f*dx*dy`10	[0] `0=0*-1*1` `f*dx*dy`0	[0] `0=0*-1*2` `f*dx*dy`0	29 `29=3+16+10+0+0`	-29 `-29=29*-1` `fdx=f_x*dx`	29 `29=-29*-1` `fdx^2=fdx*dx`	22 `22=6+16+0+0+0`
35 - 45	40 `40=(35+45)/2`	0 `0=(40-40)/10` `dx=(x-40)/10`	[0] `0=0*0*-2` `f*dx*dy`0	[0] `0=10*0*-1` `f*dx*dy`10	[0] `0=15*0*0` `f*dx*dy`15	[0] `0=7*0*1` `f*dx*dy`7	[0] `0=0*0*2` `f*dx*dy`0	32 `32=0+10+15+7+0`	0 `0=32*0` `fdx=f_x*dx`	0 `0=0*0` `fdx^2=fdx*dx`	0 `0=0+0+0+0+0`
45 - 55	50 `50=(45+55)/2`	1 `1=(50-40)/10` `dx=(x-40)/10`	[0] `0=0*1*-2` `f*dx*dy`0	[0] `0=0*1*-1` `f*dx*dy`0	[0] `0=7*1*0` `f*dx*dy`7	[10] `10=10*1*1` `f*dx*dy`10	[8] `8=4*1*2` `f*dx*dy`4	21 `21=0+0+7+10+4`	21 `21=21*1` `fdx=f_x*dx`	21 `21=21*1` `fdx^2=fdx*dx`	18 `18=0+0+0+10+8`
55 - 65	60 `60=(55+65)/2`	2 `2=(60-40)/10` `dx=(x-40)/10`	[0] `0=0*2*-2` `f*dx*dy`0	[0] `0=0*2*-1` `f*dx*dy`0	[0] `0=0*2*0` `f*dx*dy`0	[8] `8=4*2*1` `f*dx*dy`4	[20] `20=5*2*2` `f*dx*dy`5	9 `9=0+0+0+4+5`	18 `18=9*2` `fdx=f_x*dx`	36 `36=18*2` `fdx^2=fdx*dx`	28 `28=0+0+0+8+20`
		`f_y`	9 `9=6+3+0+0+0`	29 `29=3+16+10+0+0`	32 `32=0+10+15+7+0`	21 `21=0+0+7+10+4`	9 `9=0+0+0+4+5`	100 `n=sum f_x=100=9+29+32+21+9` OR `n=sum f_y=100=9+29+32+21+9`	-8 `sum fdx=-8=-18-29+0+21+18`	122 `sum fdx^2=122=36+29+0+21+36`	98 `sum fdxdy=98=30+22+0+18+28`
		`fdy`	-18 `-18=9*-2` `fdy=f_y*dy`	-29 `-29=29*-1` `fdy=f_y*dy`	0 `0=32*0` `fdy=f_y*dy`	21 `21=21*1` `fdy=f_y*dy`	18 `18=9*2` `fdy=f_y*dy`	-8 `sum fdy=-8=-18-29+0+21+18`
		`fdy^2`	36 `36=-18*-2` `fdy^2=fdy*dy`	29 `29=-29*-1` `fdy^2=fdy*dy`	0 `0=0*0` `fdy^2=fdy*dy`	21 `21=21*1` `fdy^2=fdy*dy`	36 `36=18*2` `fdy^2=fdy*dy`	122 `sum fdy^2=122=-18-29+0+21+18`
		`fdxdy`	30 `30=24+6+0+0+0`	22 `22=6+16+0+0+0`	0 `0=0+0+0+0+0`	18 `18=0+0+0+10+8`	28 `28=0+0+0+8+20`	98 `sum fdxdy=98=30+22+0+18+28`

`bar X = A + (sum fdx)/n * h_x`

`=40 + -8/100 * 10`

`=40 + -0.08 * 10`

`=40 + -0.8`

`=39.2`


`bar Y = B + (sum fdy)/n * h_y`

`=35 + -8/100 * 10`

`=35 + -0.08 * 10`

`=35 + -0.8`

`=34.2`


`byx = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdx^2 - (sum fdx)^2) * (h_y)/(h_x)`

`=(100 * 98 - -8 * -8 )/(100 * 122 - (-8)^2) * 10/10`

`=(9800 - 64 )/(12200 - 64) * 10/10`

`=9736/12136 * 10/10`

`=0.8`


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

`y - 34.2 = 0.8 (x - 39.2)`

`y - 34.2 = 0.8 x - 31.45`

`y = 0.8 x - 31.45 + 34.2`

`y = 0.8 x + 2.75`


`bxy = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdy^2 - (sum fdy)^2) * (h_x)/(h_y)`

`=(100 * 98 - -8 * -8 )/(100 * 122 - (-8)^2) * 10/10`

`=(9800 - 64 )/(12200 - 64) * 10/10`

`=9736/12136 * 10/10`

`=0.8`


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

`x - 39.2 = 0.8 (y - 34.2)`

`x - 39.2 = 0.8 y - 27.44`

`x = 0.8 y - 27.44 + 39.2`

`x = 0.8 y + 11.76`

---