Calculate Correlation Coefficient r without cov(x,y), Correlation Coefficient r with population cov(x,y), Correlation Coefficient r with sample cov(x,y) from the following data
X | Y |
300 | 800 |
350 | 900 |
400 | 1000 |
450 | 1100 |
500 | 1200 |
550 | 1300 |
600 | 1400 |
650 | 1500 |
700 | 1600 |
Solution:Mean
ˉx=∑xin=300+350+400+450+500+550+600+650+7009=45009=500Mean
ˉy=∑yin=800+900+1000+1100+1200+1300+1400+1500+16009=108009=1200x | y | X=x-50050 | Y=y-1200100 | X2 | Y2 | X⋅Y |
300 | 800 | -4 | -4 | 16 | 16 | 16 |
350 | 900 | -3 | -3 | 9 | 9 | 9 |
400 | 1000 | -2 | -2 | 4 | 4 | 4 |
450 | 1100 | -1 | -1 | 1 | 1 | 1 |
500 | 1200 | 0 | 0 | 0 | 0 | 0 |
550 | 1300 | 1 | 1 | 1 | 1 | 1 |
600 | 1400 | 2 | 2 | 4 | 4 | 4 |
650 | 1500 | 3 | 3 | 9 | 9 | 9 |
700 | 1600 | 4 | 4 | 16 | 16 | 16 |
--- | --- | --- | --- | --- | --- | --- |
4500 | 10800 | ∑X=0 | ∑Y=0 | ∑X2=60 | ∑Y2=60 | ∑X⋅Y=60 |
Correlation Coefficient r : r=∑XY√∑X2⋅√∑Y2=60√60⋅√60=607.746⋅7.746=1
Correlation Coefficient r with Population Cov(x,y) :
Population Cov(x,y)=∑(x-ˉx)(y-ˉy)n
=609
=6.6667
Population Standard deviation σ=√∑(x-ˉx)2n
=√609
=√6.6667
=2.582
Population Standard deviation σ=√∑(y-ˉy)2n
=√609
=√6.6667
=2.582
Now, r=cov(x,y)σx⋅σy
=6.66672.582⋅2.582
=1
Correlation Coefficient r with Sample Cov(x,y) :
Sample Cov(x,y)=∑(x-ˉx)(y-ˉy)n-1
=608
=7.5
Sample Standard deviation σ=√∑(x-ˉx)2n-1
=√608
=√7.5
=2.7386
Sample Standard deviation σ=√∑(y-ˉy)2n-1
=√608
=√7.5
=2.7386
Now, r=cov(x,y)σx⋅σy
=7.52.7386⋅2.7386
=1
This material is intended as a summary. Use your textbook for detail explanation.
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