2. Covariance - Population Covariance, Sample Covariance for bivariate grouped data Formula & Example-1

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2. Covariance - Population Covariance, Sample Covariance for bivariate grouped data example ( Enter your problem )

( Enter your problem )

Formula & Example-1
Example-2
Example-3

Other related methods

Correlation Coefficient r
Covariance - Population Covariance, Sample Covariance

1. Correlation Coefficient r
(Previous method)

2. Example-2
(Next example)

1. Formula & Example-1

Formula

1. `r = (n * sum fdxdy - sum fdx * sum fdy)/(sqrt(n * sum f dx^2 - (sum f dx)^2) * sqrt(n * sum f dy^2 - (sum f dy)^2))`

2. Population `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n)`

3. Sample `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n-1)`

Examples
1. Calculate Population cov(x,y), Sample cov(x,y) 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`

Population Cov(x,y) :
Population `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n)`

`=(98 - (-8 xx -8)/100)/100`

`=(98 - (64)/100)/100`

`=(98 - 0.64)/100`

`=(97.36)/100`

`=0.9736`

Sample Cov(x,y) :
Sample `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n-1)`

`=(98 - (-8 xx -8)/100)/99`

`=(98 - (64)/100)/99`

`=(98 - 0.64)/99`

`=(97.36)/99`

`=0.9834`

This material is intended as a summary. Use your textbook for detail explanation.
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