Statistics
Mean Deviation About Median — k²
nta_pyq_2026_jan
Grade 11
Question:
If the mean deviation about the median of the numbers $k,2k,3k,\ldots,1000k$ is 500, then $k^2$ is equal to:
Step-by-Step Solution
Key Concept: 1000 terms in AP. Median $=\tfrac{500k+501k}{2}=\tfrac{1001k}{2}$. $MD=\tfrac{1}{1000}\sum_{i=1}^{1000}|ik-\text{median}|$. By symmetry, sum $=2\sum_{i=1}^{500}\left(\tfrac{1001k}{2}-ik\right)=k\cdot\sum_{i=1}^{500}(1001-2i)$.
$k=2$. $k^2=4$.
Correct Answer: 3