For a given data set \x_1, x_2, , x_n\, where n = 100, it is known that 1/2000 _i=1^n _j=1^n (x_i - x_j)^2 = 99 Let us denote x = 1n _i=1^n x_i. The value of 199 _i=1^n (x_i - x)^2 is __________. (Answer in integer)

GATE DA 2026 Q62: Sample variance from the sum of squared pairwise differences

For a given data set $\{x_1, x_2, \ldots, x_n\}$ , where $n = 100$ , it is known that

$\frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99$

Let us denote $\bar{x} = \dfrac{1}{n} \sum_{i=1}^{n} x_i$ .

The value of $\dfrac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2$ is __________. (Answer in integer)

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Answer: 10 (official GATE 2026 final answer key)

Concept used: Identity: $\sum_i \sum_j (x_i - x_j)^2 = 2n \sum_i (x_i - \bar{x})^2$ .

First step: $\dfrac{2 \times 100}{2000} \sum_i (x_i - \bar{x})^2 = 99$ , so $\sum_i (x_i - \bar{x})^2 = 990$ .

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key