546. Two controllers $A$ and $B$ arranged samples of products made by nine masters in the order of decreasing quality (in parentheses are the ordinal numbers of products of the same quality):
(A)
$\begin{array}{llll}\text { (A) } & 1 & 2 \\ \text { (B) } & 2 & 1\end{array}$
$\left.\begin{array}{ccc}(3, & 4 & 5 \\ 4 & 3 & 5\end{array}\right)$
$(6$,
$7,8,9)$
$(6,7) \quad 8 \quad 9$

Find the sample rank correlation coefficient of Spearman between the ranks of the products assigned by the two controllers.
The solution is as follows. Given that the ranks of items of the same quality are equal to the arithmetic mean of the ordinal numbers of the items: $(3+4+5) / 3=4, \quad(6+7+8+9) / 4=7.5, \quad(6+7) / 2=6.5$, we can write the sequences of ranks assigned to the items by the controllers:

$$
\begin{array}{cccccccccc}
x_{i} & 1 & 2 & 4 & 4 & 4 & 7.5 & 7.5 & 7.5 & 7.5 \\
y_{i} & 2 & 1 & 4 & 3 & 5 & 6.5 & 6.5 & 8 & 9
\end{array}
$$

We will find the sample rank correlation coefficient of Spearman, given that $\sum d_{i}^{2}=8.5, n=9$:

$$
\rho_{\mathrm{B}}=1-\frac{6 \sum d_{i}^{2}}{(n-1) n(n+1)}=1-\frac{6 \cdot 8.5}{8 \cdot 9 \cdot 10}=0.93
$$

Thus, $\rho_{\mathrm{B}}=0.93$.