GATE 2022 ST – Question 56
For n paired observations used in Spearman rank correlation,x_i are distinct for i≥2 and x₁=x₂. Given Σd_i²=19.5,where d_i=rank(x_i)−rank(y_i),find the minimum possible n³−n as an integer.
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Correct answer: 60 to 60
Explanation
The official answer **60** comes from the conventional tie-corrected shortcut r=1−6[Σd_i²+(2³−2)/12]/(n³−n). The one two-way tie contributes .5,so imposing r≥−1 gives n³−n≥60,and n=4 gives 60. There is a source/key issue if ranks mean exact midranks and correlation means Pearson correlation of those ranks. For n=4,the centered x-rank sum of squares is 5−.5=4.5 and that of y ranks is at most 5;Cauchy–Schwarz bounds Σd_i² by (√4.5+√5)²<19.5. Hence n=4 cannot realize the stated data under exact ranking. Official grading is retained and the shortcut’s limitation is disclosed.