Calculate the Spearman's ranks correlation coefficient for the following data and interpret the result:
$\begin{matrix} X & 35 & 54 & 80 & 95 & 73 & 73 & 35 & 91 & 83 & 81 Y & 40 & 60 & 75 & 90 & 70 & 75 & 38 & 95 & 75 & 71 \end{matrix}$

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Solution

$\begin{matrix} X & Y & R a n k X & R a n k Y & d = R a n k X - R a n k Y & d^{2} 35 & 40 & 1 & 2 & 1 & 1 54 & 60 & 2 & 3 & 1 & 1 80 & 75 & 4 & 5 & 1 & 1 95 & 90 & 8 & 6 & 2 & 4 73 & 70 & 3.5 & 4.5 & - 1 & 1 73 & 75 & 3.5 & 5 & - 1.5 & 2.25 35 & 38 & 1 & 1 & 0 & 0 91 & 95 & 7 & 7 & 0 & 0 83 & 75 & 6 & 5 & 1 & 1 81 & 70 & 5 & 4.5 & 0.5 & 0.25 \end{matrix}$
$\sum d^{2} = 11.5$
$m_{1} =$ no. of times rank repeated $X = 2$
$m_{2} =$ no. of times rank repeated $y = 2$
$n =$ no. of ranks $= 10$
Sperman rank correlation coefficient $= r = 1 - 6 \frac{\sum d^{2} + \frac{1}{12} (m {_{1}}^{3} - m_{1}) + \frac{1}{12} (m {_{2}}^{3} - m_{2})}{n (n^{2} - 1)}$
$r = 1 - 6 \frac{(11.5) + \frac{1}{12} (2^{3} - 2) + \frac{1}{12} (2^{3} - 2)}{10 (100 - 1)} r = 1 - \frac{93}{990} = 1 - .09393 = 0.9060$

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Similar questions

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

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

X	:	10	12	8	15	20	25	40
Y	:	15	10	6	25	16	12	8

x

y

(1, 6), (2, 5), (3, 7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)

X	48	33	40	9	16	16	65	25	16	57
Y	13	13	24	6	15	4	20	9	6	19

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