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Find Correlation Coefficient {{50-55,55-60,60-65,65-70},{90-100,100-110,110-120,120-130},{{4,7,5,2},{6,10,7,4},{6,12,10,7},{3,8,6,3}}} [ Calculator, Method and examples ]

Solution:
Your problem `->` Correlation Coefficient {{50-55,55-60,60-65,65-70},{90-100,100-110,110-120,120-130},{{4,7,5,2},{6,10,7,4},{6,12,10,7},{3,8,6,3}}}


C.I.`(y)`90 - 100100 - 110110 - 120120 - 130
M.V.`(y)` 95 `95=(90+100)/2` 105 `105=(100+110)/2` 115 `115=(110+120)/2` 125 `125=(120+130)/2`
C.I.`(x)`M.V.`(x)`
`dy`
`dx`
 -1 `-1=(95-105)/10`
`dy=(y-105)/10`
 0 `0=(105-105)/10`
`dy=(y-105)/10`
 1 `1=(115-105)/10`
`dy=(y-105)/10`
 2 `2=(125-105)/10`
`dy=(y-105)/10`
`f_x``fdx``fdx^2``fdxdy`
50 - 55 52.5 `52.5=(50+55)/2` -1 `-1=(52.5-57.5)/5`
`dx=(x-57.5)/5`
 [4] `4=4*-1*-1`
`f*dx*dy`
4
 [0] `0=7*-1*0`
`f*dx*dy`
7
 [-5] `-5=5*-1*1`
`f*dx*dy`
5
 [-4] `-4=2*-1*2`
`f*dx*dy`
2
 18 `18=4+7+5+2` -18 `-18=18*-1`
`fdx=f_x*dx`
 18 `18=-18*-1`
`fdx^2=fdx*dx`
 -5 `-5=4+0-5-4`
55 - 60 57.5 `57.5=(55+60)/2` 0 `0=(57.5-57.5)/5`
`dx=(x-57.5)/5`
 [0] `0=6*0*-1`
`f*dx*dy`
6
 [0] `0=10*0*0`
`f*dx*dy`
10
 [0] `0=7*0*1`
`f*dx*dy`
7
 [0] `0=4*0*2`
`f*dx*dy`
4
 27 `27=6+10+7+4` 0 `0=27*0`
`fdx=f_x*dx`
 0 `0=0*0`
`fdx^2=fdx*dx`
 0 `0=0+0+0+0`
60 - 65 62.5 `62.5=(60+65)/2` 1 `1=(62.5-57.5)/5`
`dx=(x-57.5)/5`
 [-6] `-6=6*1*-1`
`f*dx*dy`
6
 [0] `0=12*1*0`
`f*dx*dy`
12
 [10] `10=10*1*1`
`f*dx*dy`
10
 [14] `14=7*1*2`
`f*dx*dy`
7
 35 `35=6+12+10+7` 35 `35=35*1`
`fdx=f_x*dx`
 35 `35=35*1`
`fdx^2=fdx*dx`
 18 `18=-6+0+10+14`
65 - 70 67.5 `67.5=(65+70)/2` 2 `2=(67.5-57.5)/5`
`dx=(x-57.5)/5`
 [-6] `-6=3*2*-1`
`f*dx*dy`
3
 [0] `0=8*2*0`
`f*dx*dy`
8
 [12] `12=6*2*1`
`f*dx*dy`
6
 [12] `12=3*2*2`
`f*dx*dy`
3
 20 `20=3+8+6+3` 40 `40=20*2`
`fdx=f_x*dx`
 80 `80=40*2`
`fdx^2=fdx*dx`
 18 `18=-6+0+12+12`
`f_y` 19 `19=4+6+6+3` 37 `37=7+10+12+8` 28 `28=5+7+10+6` 16 `16=2+4+7+3` 100 `n=sum f_x=100=18+27+35+20`
OR
`n=sum f_y=100=19+37+28+16`
 57 `sum fdx=57=-18+0+35+40` 133 `sum fdx^2=133=18+0+35+80` 31 `sum fdxdy=31=-5+0+18+18`
`fdy` -19 `-19=19*-1`
`fdy=f_y*dy`
 0 `0=37*0`
`fdy=f_y*dy`
 28 `28=28*1`
`fdy=f_y*dy`
 32 `32=16*2`
`fdy=f_y*dy`
 41 `sum fdy=41=-19+0+28+32`
`fdy^2` 19 `19=-19*-1`
`fdy^2=fdy*dy`
 0 `0=0*0`
`fdy^2=fdy*dy`
 28 `28=28*1`
`fdy^2=fdy*dy`
 64 `64=32*2`
`fdy^2=fdy*dy`
 111 `sum fdy^2=111=-19+0+28+32`
`fdxdy` -8 `-8=4+0-6-6` 0 `0=0+0+0+0` 17 `17=-5+0+10+12` 22 `22=-4+0+14+12` 31 `sum fdxdy=31=-8+0+17+22`


Correlation Coefficient r :
`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))`

`=(100 * 31 - 57 * 41 )/(sqrt(100 * 133 - (57)^2) * sqrt(100 * 111 - (41)^2)`

`=(3100 - 2337)/(sqrt(13300 - 3249) * sqrt(11100 - 1681))`

`=763/( sqrt(10051) * sqrt(9419))`

`=763/( 100.2547 * 97.0515)`

`=763/9729.8699`

`=0.0784`



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

`=(31 - (57 xx 41)/100)/100`

`=(31 - (2337)/100)/100`

`=(31 - 23.37)/100`

`=(7.63)/100`

`=0.0763`


Population Standard deviation `sigma_x = sqrt((sum fdx^2 - (sum fdx)^2/n)/(n))`

`=sqrt((133 - (57)^2/100)/100)`

`=sqrt((133 - 32.49)/100)`

`=sqrt(100.51/100)`

`=sqrt(1.0051)`

`=1.0025`

Population Standard deviation `sigma_y = sqrt((sum fdy^2 - (sum fdy)^2/n)/(n))`

`=sqrt((111 - (41)^2/100)/100)`

`=sqrt((111 - 16.81)/100)`

`=sqrt(94.19/100)`






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