Question

The marks obtained by nine students in physics and Mathematics are given below:

Physics $:48,60,72,62,56,40,39,52,30$

Mathematics$:62,78,65,70,38,54,60,32,31$.

Calculate Spearman's coefficient.

Answer 1

Step 1: Arranging the data in descending order to find the rank:

On arranging the given data in descending order,

Physics$:72,62,60,56,52,48,40,39,30$

Mathematics$:78,70,65,62,60,54,38,32,31$

Here, the number of students, $n=9$.

The highest value is ranked as one and the lowest as the last i.e. $9$.

Step 2: Completing the table to find values of $\left|d\right|=\left|Rank\left(P\right)-Rank\left(M\right)\right|$, $d^2$:

Physics(P)	Mathematics(M)	Rank (P)	Rank (M)	$\left|d\right|=\left|Rank\left(P\right)-Rank\left(M\right)\right|$	$d^2$
$48$	$62$	$6$	$4$	$2$	$4$
$60$	$78$	$3$	$1$	$2$	$4$
$72$	$65$	$1$	$3$	$2$	$4$
$62$	$70$	$2$	$2$	$0$	$0$
$56$	$38$	$4$	$7$	$3$	$9$
$40$	$54$	$7$	$6$	$1$	$1$
$39$	$60$	$8$	$5$	$3$	$9$
$52$	$32$	$5$	$8$	$3$	$9$
$30$	$31$	$9$	$9$	$0$	$0$

Step 3: Calculating Spearman's coefficient:

Spearman's coefficient, $\mathrm{r}=1-\frac{6\sum {\mathrm{d}}^2}{\mathrm{n}({\mathrm{n}}^2-1)}$

$$\begin{gathered} \sum {\mathrm{d}}^2=4+4+4+0+9+1+9+9+0=40 \\ \therefore \mathrm{r}=1-\frac{6\sum {\mathrm{d}}^2}{\mathrm{n}({\mathrm{n}}^2-1)} \\ =1-\frac{40\times 6}{9(9^2-1)}=1-\frac{240}{720}=0.66 \end{gathered}$$

Thus, the value of spearman's coefficient is $0.66$.