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Question
Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result:
| X | 35 | 54 | 80 | 95 | 73 | 73 | 35 | 91 | 83 | 81 |
| Y | 40 | 60 | 75 | 90 | 70 | 75 | 38 | 95 | 75 | 70 |
Solution
To determine Spearman’s Rank Correlation:
| X | Y | R1 | R2 | d = R1 - R2 | d2 |
| 35 | 40 | 9.5 | 9 | 0.5 | 0.25 |
| 54 | 60 | 8 | 8 | 0 | 0 |
| 80 | 75 | 5 | 4 | 1 | 1 |
| 95 | 90 | 1 | 2 | -1 | 1 |
| 73 | 70 | 6.5 | 6.5 | 0 | 0 |
| 73 | 75 | 6.5 | 4 | 2.5 | 6.25 |
| 35 | 38 | 9.5 | 10 | -0.5 | 0.25 |
| 91 | 95 | 2 | 1 | 1 | 1 |
| 83 | 75 | 3 | 4 | -1 | 1 |
| 81 | 70 | 4 | 6.5 | -2.5 | 6.25 |
| Total | ∑d2 = 17 |
r = 1 - `(6[sum"d"^2 + sum (("t"^3 - "t")/12)])/("N"("N"^2 - 1))`
where t is the number of individual involved in a tie.
Here, ∑d2 = 17, N = 10 , t = 3,2,2,2
`sum("t"^3-"t")/12 = ((3)^2 - 3)/12 + 3 xx ((2)^3 - 2)/12`
`= (27-3)/12 + 3 xx (8 - 2)/12`
`= 24/12 + 18/12 = 42/12 =7/2 = 3.5`
r = `1 - (6[17 + 3.5])/(10(100 - 1))`
`=1 - 123/990`
`= (990-123)/990`
`= 867/990 = 0.876`
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