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Problem: Correlation Coefficient {{10,3},{11,12},{12,18},{13,12},{14,3}} [ Calculator, Method and examples ]

Solution:
Your problem `->` Correlation Coefficient {{10,3},{11,12},{12,18},{13,12},{14,3}}


`x``y``x^2``y^2``x*y`
103100930
1112121144132
1218144324216
1312169144156
143196942
---------------
`sum x=60``sum y=48``sum x^2=730``sum y^2=630``sum xy=576`


Correlation Coefficient r :
`r = (n * sum xy - sum x * sum y)/(sqrt(n * sum x^2 - (sum x)^2) * sqrt(n * sum y^2 - (sum y)^2))`

`=(5 * 576 - 60 * 48 )/(sqrt(5 * 730 - (60)^2) * sqrt(5 * 630 - (48)^2)`

`=(2880 - 2880)/(sqrt(3650 - 3600) * sqrt(3150 - 2304))`

`=0/( sqrt(50) * sqrt(846))`

`=0/( 7.0711 * 29.0861)`

`=0/205.6696`

`=0`



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

`=(576 - (60 xx 48)/5)/5`

`=(576 - (2880)/5)/5`

`=(576 - 576)/5`

`=(0)/5`

`=0`


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

`=sqrt((730 - (60)^2/5)/5)`

`=sqrt((730 - 720)/5)`

`=sqrt(10/5)`

`=sqrt(2)`

`=1.4142`

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

`=sqrt((630 - (48)^2/5)/5)`

`=sqrt((630 - 460.8)/5)`

`=sqrt(169.2/5)`






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