Question

While calculating Spearman’s rank correlation coefficient on $n$ paired observations, it is found that $x_i$ are distinct for all $i\ge 2$, with a single tie $x_1=x_2$, and $\sum_{i=1}^{n} d_i^2=19.5$ where $d_i=\text{rank}(x_i)-\text{rank}(y_i)$. Then the minimum possible value of $\,n^3-n\,$ (in integer) is $\;__________$.

Hint:

Spearman’s $\sum d_i^2$ is maximized (and thus bounded) by $n(n^2-1)/3$. Use this bound to find the smallest $n$ compatible with a given $\sum d_i^2$.

Answer 1

Correct Answer: 60

For ranks without considering tie corrections, the well–known upper bound is \[ \sum_{i=1}^{n} d_i^2 \;\le\; \frac{n(n^2-1)}{3}, \] (attained when the two rankings are in reverse order). With only one tie in $x$ the attainable values may be half–integers, but the same upper bound still controls the sum. We need the smallest $n$ such that $19.5\le \dfrac{n(n^2-1)}{3}$. Check $n=3$: $\dfrac{3(9-1)}{3}=8<19.5$. For $n=4$: $\dfrac{4(16-1)}{3}=20\ge 19.5$ (feasible due to the half–integer total from the single tie). Hence the minimum $n$ is $4$, and therefore \[ n^3-n=4^3-4=64-4=\boxed{60}. \]