Question

The candidates received percentage marks in two subjects as follows

Candidate	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Mathematics	$80$	$88$	$76$	$74$	$68$	$65$	$40$	$43$	$40$	$80$
Statastics Marks	$72$	$84$	$90$	$66$	$54$	$50$	$54$	$38$	$30$	$43$

Calculate Spearman's rank correlation coefficient and interpret your result.

Answer 1

Candidate	Marks (Maths)	$R_1$	Marks Stat.	$R_2$	$d=R_1-R_2$	$d^2$
$A$	$80$	$2.5$	$72$	$3$	$-0.5$	$0.25$
$B$	$88$	$1$	$84$	$2$	$-1$	$01$
$C$	$76$	$4$	$90$	$1$	$3$	$09$
$D$	$74$	$5$	$66$	$4$	$1$	$01$
$E$	$68$	$6$	$54$	$5.5$	$0.5$	$0.25$
$F$	$65$	$7$	$50$	$7$	$0$	$0$
$G$	$40$	$9.5$	$54$	$5.5$	$4$	$16$
$H$	$43$	$8$	$38$	$9$	$-1$	$01$
$I$	$40$	$9.5$	$30$	$10$	$-0.5$	$0.25$
$J$	$80$	$2.5$	$43$	$8$	$-5.5$	$30.25$
						$\sum d^2=59$

There are two ties in $R_1$ each of two items. Also there is one tie in $R_2$ of two items.
$r=1-\frac{6[\sum d^2+\frac{1}{12}(m^3-m\left)\right]}{n(n^2-1)}$

$=1-\frac{[59+\frac{1}{12}(2^3-2)+\frac{1}{12}(2^3-2)+\frac{1}{12}(2^3-2\left)\right]}{10(100-1)}$

$=1-\frac{6[59+1.5]}{10\times 99}=1-\frac{6\times 60.5}{990}$

$=1-0.367=0.633$

This result indicates a good positive relationship between mathematics and statistics marks.