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Home > Statistical Methods calculators > Regression line equations for bivariate grouped data example
Regression line equations for bivariate grouped data example ( Enter your problem )
( Enter your problem )
  1. Example-1 (Class-X & Class-Y)
  2. Example-2 (Class-X & Y)
  3. Example-3 (Class-X & Class-Y)
2. Example-2 (Class-X & Y)
(Next example)

1. Example-1 (Class-X & Class-Y)


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 - 2020 - 3030 - 4040 - 5050 - 60
15 - 2563000
25 - 353161000
35 - 450101570
45 - 55007104
55 - 6500045


Solution:
C.I.`(y)`10 - 2020 - 3030 - 4040 - 5050 - 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`

This material is intended as a summary. Use your textbook for detail explanation.
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2. Example-2 (Class-X & Y)
(Next example)


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