Calculate Spearman's rank correlation coefficient for the given ranks.

Question

ScanSolve AI Tutor · Accepted Answer

To find Spearman's rank correlation coefficient, we use the formula:

\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}

where $n$ is the number of items and $d_i$ is the difference between the ranks of each item.

The true order (true ranks) is 1, 2, 3, 4.
The observed order (observed ranks) is 2, 3, 1, 4.
The number of items, $n = 4$ .

Item	True Rank ( $R_i$ )	Observed Rank ( $S_i$ )	$d_i = R_i - S_i$	$d_i^2$
1	1	2	$1 - 2 = -1$	1
2	2	3	$2 - 3 = -1$	1
3	3	1	$3 - 1 = 2$	4
4	4	4	$4 - 4 = 0$	0

Calculate the sum of the squared differences, $\sum d_i^2$ .
$\sum d_i^2 = 1 + 1 + 4 + 0 = 6$
Substitute the values of $n$ and $\sum d_i^2$ into Spearman's rank correlation coefficient formula.
$\rho = 1 - \frac{6 \times 6}{4(4^2 - 1)}$ $\rho = 1 - \frac{36}{4(16 - 1)}$ $\rho = 1 - \frac{36}{4(15)}$ $\rho = 1 - \frac{36}{60}$
Simplify the expression to find $\rho$ .
$\rho = 1 - \frac{3}{5}$ $\rho = 1 - 0.6$ $\rho = 0.4$
The value of Spearman's rank correlation coefficient is $\text{0.4}$ .

Answer