Question

Find out rank difference correlation of X and Y:

X	80	78	75	75	58	67	60	59
Y	12	13	14	14	14	16	15	17

Answer 1

X	R1	Y	R2	D = R1 – R2	D2
80 78 75 75 58 67 60 59	1 2 3.5 3.5 8 5 6 7	12 13 14 14 14 16 15 17	8 7 5 5 5 2 3 1	–7 –5 –1.5 –1.5 3 3 3 6	49 25 2.25 2.25 9 9 9 36
N = 8					${\sum }_{}{D}^2=141.5$

$$\begin{gathered} r_k=1-\frac{6\left[\sum D^2+{\displaystyle \frac{1}{12}}\left(M_1^3-M_1\right)+{\displaystyle \frac{1}{12}}\left(M_2^3-M_2\right)\right]}{N^3-N} \\ \mathrm{Here,} \\ {M}_1=\mathrm{2,}\mathrm{as}\mathrm{the}\mathrm{item}75\mathrm{is}\mathrm{appearing}\mathrm{twice}\mathrm{in}X-\mathrm{series}. \\ {M}_2=3,\mathrm{as}\mathrm{the}\mathrm{item}14\mathrm{is}\mathrm{appearing}\mathrm{thrice}\mathrm{in}Y-\mathrm{series}. \\ \mathrm{or},r_k=1-\frac{6\left[141.50+{\displaystyle \frac{1}{12}}(2^3-2)+{\displaystyle \frac{1}{12}}(3^3-3)\right]}{512-8} \\ \therefore ,r_k=\frac{504-864}{504}=-0.714 \end{gathered}$$