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Question
Calculate Karl Pearson’s coefficient of correlation between x and y for the following data and interpret the result:
(1, 6), (2, 5), (3, 7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)
Solution
Here n = 9, `barx = (45)/(9) = 5, bary = (81)/(9) = 9`
| x | y | `x - barx` | `y - bary` | `(x - barx)^2` | `(y - bary)^2` | `(x - barx) (y - bary` |
| 1 | 6 | - 4 | - 3 | 16 | 9 | 12 |
| 2 | 5 | - 3 | - 4 | 9 | 16 | 12 |
| 3 | 7 | - 2 | - 2 | 4 | 4 | 4 |
| 4 | 9 | - 1 | 0 | 1 | 0 | 0 |
| 5 | 8 | 0 | - 1 | 0 | 1 | 0 |
| 6 | 10 | 1 | 1 | 1 | 1 | 1 |
| 7 | 11 | 2 | 2 | 4 | 4 | 4 |
| 8 | 13 | 3 | 3 | 9 | 16 | 12 |
| 9 | 12 | 4 | 4 | 16 | 9 | 12 |
| ∑x = 45 | ∑y = 81 | ∑`(x -barx)^2` = 60 | ∑`(y -bary)^2` = 60 | ∑`(x -barx) (y -bary)` = 57 |
r = `(∑(x - barx) (y - bary))/(sqrt(∑(x - barx)^2) sqrt(∑(y -bary)^2)) = 57/(sqrt60 sqrt60) = (57)/(60) = 0.95`
| x | y | u = x - 17 | v = y - 19 | u2 | v2 | uv |
| 20 | 17 | 3 | - 2 | 9 | 4 | - 6 |
| 13 | 12 | - 4 | - 7 | 16 | 49 | 28 |
| 18 | 23 | 1 | 4 | 1 | 16 | 4 |
| 21 | 25 | 4 | 6 | 16 | 36 | 24 |
| 11 | 14 | - 6 | - 5 | 36 | 25 | 30 |
| 12 | 8 | - 5 | - 11 | 25 | 121 | 55 |
| 17 | 19 | 0 | 0 | 0 | 0 | 0 |
| 14 | 21 | - 3 | 2 | 9 | 4 | - 6 |
| 19 | 22 | 2 | 3 | 4 | 9 | 6 |
| 15 | 19 | - 2 | 0 | 4 | 0 | 0 |
| ∑u = -10 | ∑v = -10 | ∑u2 = 120 | ∑v2 = 264 | ∑uv = 135 |
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