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In Spearman rank correlation coefficient rs = 1 - \(\dfrac{6∑ d^2}{n(n^2-1)}\), the maximum value of ∑d2 in case of untied rank is:
1. \(\dfrac{1}{2}(n^2 - 1)\)
2. \(\dfrac{1}{4}n(n^2 - 1)\)
3. n
4. \(\dfrac{1}{3}n(n^2 - 1)\)

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Correct Answer - Option 4 : \(\dfrac{1}{3}n(n^2 - 1)\)

Given

In the case of the untied rank maximum value of r= -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 ∑din case of untied rank is = \(\dfrac{1}{3}n(n^2 - 1)\)

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