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Question
Calculate the Karl Pearson Correlation Co-efficient for the following data:
| Demand for Product X: | 23 | 27 | 28 | 29 |
30 |
31 | 33 | 35 | 36 | 39 |
| Sale of Product Y: | 18 | 22 | 23 | 24 | 25 | 26 | 28 | 29 | 30 | 32 |
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Solution
| Sr. No. | X | Y | (X-A) = dx | (Y-A) = dy | dx2 | dy2 | dxdy |
| 1 | 23 | 18 | −8 | −8 | 64 | 64 | 64 |
| 2 | 27 | 22 | −4 | −4 | 16 | 16 | 16 |
| 3 | 28 | 23 | −3 | −3 | 9 | 9 | 9 |
| 4 | 29 | 24 | −2 | −2 | 4 | 4 | 4 |
| 5 | 30 | 25 | −1 | −1 | 1 | 1 | 1 |
| 6 | 31 | 26 | 0 | 0 | 0 | 0 | 0 |
| 7 | 33 | 28 | 2 | 2 | 4 | 4 | 4 |
| 8 | 35 | 29 | 4 | 3 | 16 | 9 | 12 |
| 9 | 36 | 30 | 5 | 4 | 25 | 16 | 20 |
| 10 | 39 | 32 | 8 | 6 | 64 | 36 | 48 |
| N = 10 | ∑X = 311 | ∑Y = 257 | ∑(X−A) = 1 | ∑(Y-A) = (−2) | ∑dx2 = 203 | ∑dy2 = 159 | ∑dxdy = 178 |
`barx = (sumX)/N = 311/10 = 31.1`
`barx = (sumY)/N = 257/10 = 25.7`
Take the assumed values A = 31 and B = 26
Therefore
dx = X − A ⇒ X − 31 and
dy = Y − A ⇒ Y − 26
`∴ r = (Nsumdxdy - (sumdx)(sumdy))/(sqrt(Nsumdx^2-(sumdx)^2)sqrt(Nsumdy^2-(sumdy)^2)`
`= (10xx178 -1xx(-2))/(sqrt(10xx203- (1)^2) xx sqrt(10xx159 -(-3)^2)`
= `r = (1780 + 2)/(sqrt(2030 - 1) * sqrt(1590 - 4)) = (1782)/(sqrt(2029*1586))`
= `r = (1782)/(sqrt(3219494)) = (1782)/(1793.17)`
r ≈ 0.9955
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