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 `bar x = (sum x_i)/n`
`=(300+350+400+450+500+550+600+650+700)/9`
`=4500/9`
`=500`
Mean `bar y = (sum y_i)/n`
`=(800+900+1000+1100+1200+1300+1400+1500+1600)/9`
`=10800/9`
`=1200`
| `x` | `y` | `X=(x-500)/50` | `Y=(y-1200)/100` | `X^2` | `Y^2` | `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` | `sum X=0` | `sum Y=0` | `sum X^2=60` | `sum Y^2=60` | `sum X*Y=60` |
Correlation Coefficient r :
`r = (sum XY)/(sqrt(sum X^2) * sqrt(sum Y^2))`
`=60/(sqrt(60) * sqrt(60))`
`=60/(7.746 * 7.746)`
`=1`
Correlation Coefficient r with Population Cov(x,y) :
Population `Cov(x,y) = (sum (x-bar x)(y-bar y))/(n)`
`=60/9`
`=6.6667`
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/(n))`
`=sqrt(60/9)`
`=sqrt(6.6667)`
`=2.582`
Population Standard deviation `sigma = sqrt((sum (y - bar y)^2)/(n))`
`=sqrt(60/9)`
`=sqrt(6.6667)`
`=2.582`
Now, `r = (cov(x,y))/(sigma_x * sigma_y)`
`= (6.6667)/(2.582 * 2.582)`
`=1`
Correlation Coefficient r with Sample Cov(x,y) :
Sample `Cov(x,y) = (sum (x-bar x)(y-bar y))/(n-1)`
`=60/8`
`=7.5`
Sample Standard deviation `sigma = sqrt((sum (x - bar x)^2)/(n-1))`
`=sqrt(60/8)`
`=sqrt(7.5)`
`=2.7386`
Sample Standard deviation `sigma = sqrt((sum (y - bar y)^2)/(n-1))`
`=sqrt(60/8)`
`=sqrt(7.5)`
`=2.7386`
Now, `r = (cov(x,y))/(sigma_x * sigma_y)`
`= (7.5)/(2.7386 * 2.7386)`
`=1`
This material is intended as a summary. Use your textbook for detail explanation.
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