If the mean deviation about the median of the numbers k , 2 k , 3 k , ldots, 1000 k is 500 , then k 2 is equal to :

Mathematics · Statistics

JEE Main 2026 — 22 January, Evening Shift — Question 2

If the mean deviation about the median of the numbers $\mathrm{k}, 2 \mathrm{k}, 3 \mathrm{k}, \ldots, 1000 \mathrm{k}$ is $500$ , then $\mathrm{k}^{2}$ is equal to :

Option A:
$16$
Option B:
$4$
Correct
Option C:
$1$
Option D:
$9$

Answer: B

Step-by-step solution

$\because$ median $=\frac{1001 \mathrm{k}}{2}=\mathrm{X}_{\mathrm{M}}$

∴ mean deviation about median $=\frac{\sum\left|\mathrm{X}_{\mathrm{i}}-\mathrm{X}_{\mathrm{M}}\right|}{\mathrm{n}}$ $=\frac{2\left(\frac{\mathrm{k}}{2}+\frac{3 \mathrm{k}}{2}+\frac{5 \mathrm{k}}{2}+\ldots 500 \text { terms }\right)}{1000}$

$=\frac{2 \cdot \frac{\mathrm{k}}{2}(500)^{2}}{1000}=\frac{500 \mathrm{k}}{2}=500$ (given)

$\therefore \mathrm{k}=2$

$\therefore \mathrm{k}^{2}=4$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 22 January, Evening Shift
Subject: Mathematics
Chapter: Statistics
Topic: Measures of Dispersion