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

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Home > Statistical Methods calculators > Covariance - Population Covariance, Sample Covariance for bivariate grouped data example

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

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Correlation Coefficient r
Covariance - Population Covariance, Sample Covariance

1. Formula & Example-1
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2. Example-2

2. Calculate Population cov(x,y), Sample cov(x,y) from the following data
Y
Class-X 18 19 20 21
350 - 400 1 4 6 10
300 - 350 2 6 8 5
250 - 300 3 5 4 2
200 - 250 4 4 2 1

Solution:


		M.V.`(y)`	18	19	20	21
C.I.`(x)`	M.V.`(x)`	`dy` `dx`	-1 `-1=18-19` `dy=y-19`	0 `0=19-19` `dy=y-19`	1 `1=20-19` `dy=y-19`	2 `2=21-19` `dy=y-19`	`f_x`	`fdx`	`fdx^2`	`fdxdy`
350 - 400	375 `375=(350+400)/2`	1 `1=(375-325)/50` `dx=(x-325)/50`	[-1] `-1=11-1` `fdxdy`1	[0] `0=410` `fdxdy`4	[6] `6=611` `fdxdy`6	[20] `20=1012` `fdxdy`10	21 `21=1+4+6+10`	21 `21=211` `fdx=f_xdx`	21 `21=211` `fdx^2=fdxdx`	25 `25=-1+0+6+20`
300 - 350	325 `325=(300+350)/2`	0 `0=(325-325)/50` `dx=(x-325)/50`	[0] `0=20-1` `fdxdy`2	[0] `0=600` `fdxdy`6	[0] `0=801` `fdxdy`8	[0] `0=502` `fdxdy`5	21 `21=2+6+8+5`	0 `0=210` `fdx=f_xdx`	0 `0=00` `fdx^2=fdxdx`	0 `0=0+0+0+0`
250 - 300	275 `275=(250+300)/2`	-1 `-1=(275-325)/50` `dx=(x-325)/50`	[3] `3=3-1-1` `fdxdy`3	[0] `0=5-10` `fdxdy`5	[-4] `-4=4-11` `fdxdy`4	[-4] `-4=2-12` `fdxdy`2	14 `14=3+5+4+2`	-14 `-14=14-1` `fdx=f_xdx`	14 `14=-14-1` `fdx^2=fdxdx`	-5 `-5=3+0-4-4`
200 - 250	225 `225=(200+250)/2`	-2 `-2=(225-325)/50` `dx=(x-325)/50`	[8] `8=4-2-1` `fdxdy`4	[0] `0=4-20` `fdxdy`4	[-4] `-4=2-21` `fdxdy`2	[-4] `-4=1-22` `fdxdy`1	11 `11=4+4+2+1`	-22 `-22=11-2` `fdx=f_xdx`	44 `44=-22-2` `fdx^2=fdxdx`	0 `0=8+0-4-4`
		`f_y`	10 `10=1+2+3+4`	19 `19=4+6+5+4`	20 `20=6+8+4+2`	18 `18=10+5+2+1`	67 `n=sum f_x=67=21+21+14+11` OR `n=sum f_y=67=10+19+20+18`	-15 `sum fdx=-15=21+0-14-22`	79 `sum fdx^2=79=21+0+14+44`	20 `sum fdxdy=20=25+0-5+0`
		`fdy`	-10 `-10=10-1` `fdy=f_ydy`	0 `0=190` `fdy=f_ydy`	20 `20=201` `fdy=f_ydy`	36 `36=182` `fdy=f_ydy`	46 `sum fdy=46=-10+0+20+36`
		`fdy^2`	10 `10=-10-1` `fdy^2=fdydy`	0 `0=00` `fdy^2=fdydy`	20 `20=201` `fdy^2=fdydy`	72 `72=362` `fdy^2=fdydy`	102 `sum fdy^2=102=-10+0+20+36`
		`fdxdy`	10 `10=-1+0+3+8`	0 `0=0+0+0+0`	-2 `-2=6+0-4-4`	12 `12=20+0-4-4`	20 `sum fdxdy=20=10+0-2+12`

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

`=(20 - (-15 xx 46)/67)/67`

`=(20 - (-690)/67)/67`

`=(20 - -10.2985)/67`

`=(30.2985)/67`

`=0.4522`

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

`=(20 - (-15 xx 46)/67)/66`

`=(20 - (-690)/67)/66`

`=(20 - -10.2985)/66`

`=(30.2985)/66`

`=0.4591`

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