Regression Lines for bivariate grouped data Example-1 (Class-X & Class-Y)

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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 )

Example-1 (Class-X & Class-Y)
Example-2 (Class-X & Y)
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 - 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` `fdxdy`6	[6] `6=3-2-1` `fdxdy`3	[0] `0=0-20` `fdxdy`0	[0] `0=0-21` `fdxdy`0	[0] `0=0-22` `fdxdy`0	9 `9=6+3+0+0+0`	-18 `-18=9-2` `fdx=f_xdx`	36 `36=-18-2` `fdx^2=fdxdx`	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` `fdxdy`3	[16] `16=16-1-1` `fdxdy`16	[0] `0=10-10` `fdxdy`10	[0] `0=0-11` `fdxdy`0	[0] `0=0-12` `fdxdy`0	29 `29=3+16+10+0+0`	-29 `-29=29-1` `fdx=f_xdx`	29 `29=-29-1` `fdx^2=fdxdx`	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=00-2` `fdxdy`0	[0] `0=100-1` `fdxdy`10	[0] `0=1500` `fdxdy`15	[0] `0=701` `fdxdy`7	[0] `0=002` `fdxdy`0	32 `32=0+10+15+7+0`	0 `0=320` `fdx=f_xdx`	0 `0=00` `fdx^2=fdxdx`	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=01-2` `fdxdy`0	[0] `0=01-1` `fdxdy`0	[0] `0=710` `fdxdy`7	[10] `10=1011` `fdxdy`10	[8] `8=412` `fdxdy`4	21 `21=0+0+7+10+4`	21 `21=211` `fdx=f_xdx`	21 `21=211` `fdx^2=fdxdx`	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=02-2` `fdxdy`0	[0] `0=02-1` `fdxdy`0	[0] `0=020` `fdxdy`0	[8] `8=421` `fdxdy`4	[20] `20=522` `fdxdy`5	9 `9=0+0+0+4+5`	18 `18=92` `fdx=f_xdx`	36 `36=182` `fdx^2=fdxdx`	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_ydy`	-29 `-29=29-1` `fdy=f_ydy`	0 `0=320` `fdy=f_ydy`	21 `21=211` `fdy=f_ydy`	18 `18=92` `fdy=f_ydy`	-8 `sum fdy=-8=-18-29+0+21+18`
		`fdy^2`	36 `36=-18-2` `fdy^2=fdydy`	29 `29=-29-1` `fdy^2=fdydy`	0 `0=00` `fdy^2=fdydy`	21 `21=211` `fdy^2=fdydy`	36 `36=182` `fdy^2=fdydy`	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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