Calculate rank coefficient of correlation of the following data.

X : 80 78 75 75 68 67 60 59

Y : 12 13 14 14 14 16 15 17

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Solution

X R₁ Y R₂ (R₁ − R₂)
D D²

80
78
75
75
68
67
60
59 8
7
5.5
5.5
4
3
2
1 12
13
14
14
14
16
15
17 1
2
4
4
4
7
6
8
7
5
1.5
1.5
0
$-$ 4
$-$ 4
$-$ 7
49
25
2.25
2.25
0
16
16
49

$\sum D^{2} = 159.5$

Given:
N = 8
Now, in the given case, where a rank is repeated more than once, the following formula will be used,
$r_{k} = 1 - \frac{6 [Σ D^{2} + \frac{1}{12} (M_{1}^{3} - M_{1}) + \frac{1}{12} (M_{2}^{3} - M_{2})]}{N^{3} - N} = 1 - \frac{6 [159.5 + \frac{1}{12} (2^{3} - 2) + \frac{1}{12} (3^{3} - 3)]}{512 - 8} = 1 - \frac{6 [159.5 + 0.5 + 2]}{504} = 1 - \frac{972}{504} = 1 - 1.928 = - 0.928 or - 0.93 (approx .)$
Thus, the rank coefficient of correlation is $-$ 0.93.

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Calculate the coefficient of correlation of the following data by Spearman's rank correlation method:

$\begin{matrix} X & 19 & 24 & 12 & 23 & 19 & 16 Y & 9 & 22 & 20 & 14 & 22 & 18 \end{matrix}$

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