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Question #59917

Difficulty: 8/10 | 🟢 Free

The Question

Let $x_1, x_2, \dots, x_n$ be $n$ real observations with mean $\bar{x}$ and variance $\sigma^2$ . A new variable $u_i$ is defined as $u_i = \frac{x_i - [\bar{x}]}{h}$ for $i = 1, 2, \dots, n$ , where $h > 0$ and $[\cdot]$ denotes the greatest integer function. If $\{\cdot\}$ denotes the fractional part function, which of the following represents the sum of squares of the new observations, $\sum_{i=1}^n u_i^2$ ?

Options

A.
$\frac{n}{h^2} \left( \sigma^2 + \{\bar{x}\}^2 \right)$
B.
$\frac{n}{h^2} \sigma^2$
C.
$\frac{n}{h^2} \left( \sigma^2 - \{\bar{x}\}^2 \right)$
D.
$\frac{n}{h^2} \left( \sigma^2 + [\bar{x}]^2 \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Variance Invariance under Translation|Step-Deviation Method (u = (x-a)/h)|Integral Part and Fractional Part

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the precise application of the step-deviation method and variance translation invariance.

Students often incorrectly assume that shifting the origin by an approximate mean (like $[\bar{x}]$ ) simplifies the sum of squared deviations exactly to the variance (scaled by $n/h^2$ ), leading them to choose option B.

This trap exposes the failure to account for the non-zero mean of the shifted data ( $\bar{u} \neq 0$ ), requiring the inclusion of the fractional part $\{\bar{x}\}$ in the fundamental expansion $E[U^2] = \text{Var}(U) + (E[U])^2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free