Question

Seven students of a class secured following marks in Economics and History. Calculate coefficient of correlation with the help of these data.

Economics	66	90	89	55	58	44	42
History	58	76	65	58	53	49	56

Answer 1

Economics (E)	R1	History (H)	R2	D = R1 – R2	D2
66 90 89 55 58 44 42	3 1 2 5 4 6 7	58 76 65 58 53 49 56	3.5 1 2 3.5 6 7 5	–.5 0 0 1.5 –2 –1 2	.25 0 0 2.25 4 1 4
N = 7					$\sum D^2=11.50$

Here, note that for the marks scored in History, two ranks are tied. That is, two students scored 58 marks. Thus, we use the following formula for the calculation of correlation.
$$\begin{gathered} r_k=1-\frac{6\left[\sum D^2+{\displaystyle \frac{1}{12}}\left(M_1^3-M_1\right)\right]}{N^3-N} \\ \mathrm{Here,}{M}_1=\mathrm{2,}\mathrm{as}\mathrm{two}\mathrm{students}\mathrm{have}\mathrm{scored}58\mathrm{in}\mathrm{History}. \\ \mathrm{or},r_k=1-\frac{6\left[11.50+{\displaystyle \frac{1}{12}}(8-2)\right]}{343-7} \\ \therefore ,r_k=\frac{336-72}{336}=+0.79 \end{gathered}$$
Thus, there exists a positive correlation between marks scored in Economics and marks scored in History.