Calculate the rank correlation coefficient using Spearman's formula, given the ranks of two methods.

Question

ScanSolve AI Tutor · Accepted Answer

To calculate the rank correlation coefficient, we will use Spearman's Rank Correlation Coefficient formula.

Let $R_A$ be the rank by students of A and $R_B$ be the rank by students of B.

Method	$R_A$	$R_B$	$d = R_A - R_B$	$d^2$
I	2	1	$2 - 1 = 1$	$1^2 = 1$
II	1	3	$1 - 3 = -2$	$(-2)^2 = 4$
III	3	2	$3 - 2 = 1$	$1^2 = 1$
IV	5	4	$5 - 4 = 1$	$1^2 = 1$
V	4	7	$4 - 7 = -3$	$(-3)^2 = 9$
VI	6	5	$6 - 5 = 1$	$1^2 = 1$
VII	7	6	$7 - 6 = 1$	$1^2 = 1$

Sum the squared differences.
$\sum d^2 = 1 + 4 + 1 + 1 + 9 + 1 + 1 = 18$
Identify the number of pairs of observations ( $n$ ).
There are 7 methods, so $n = 7$ .
Apply Spearman's Rank Correlation Coefficient formula.
The formula is:
$\rho = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}$
Substitute the values:
$\rho = 1 - \frac{6 \times 18}{7(7^2 - 1)}$ $\rho = 1 - \frac{108}{7(49 - 1)}$ $\rho = 1 - \frac{108}{7(48)}$ $\rho = 1 - \frac{108}{336}$
Simplify the fraction $\frac{108}{336}$ by dividing both numerator and denominator by their greatest common divisor, which is 12:
$\rho = 1 - \frac{108 \div 12}{336 \div 12}$ $\rho = 1 - \frac{9}{28}$
To subtract, find a common denominator:
$\rho = \frac{28}{28} - \frac{9}{28}$ $\rho = \frac{19}{28}$
Convert to a decimal:
$\rho \approx 0.67857$
The rank correlation coefficient is 0.6786.

Answer