Correct Answer - Option 4 :
\(\dfrac{1}{3}n(n^2 - 1)\)
Given
In the case of the untied rank maximum value of rs = -1
Calculation
Here, Spearman rank correlation coefficient rs = -1
⇒ 1 - \(\dfrac{6∑ d^2}{n(n^2-1)}\) = -1
⇒ \(\dfrac{6∑ d^2}{n(n^2-1)}\) = 2
⇒ 6 \(∑ \)d2 = 2n(n2 - 1)
⇒ Σd2 = n(n2 - 1)/3
∴ The maximum value of ∑d2 in case of untied rank is = \(\dfrac{1}{3}n(n^2 - 1)\)