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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}}} [ Calculator, Method and examples ]

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

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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