Statistics
Statistics
nta_pyq_2025_jan
Grade 11
Question:
The variance of the numbers $8,21,34,47,\dots,320$ is \rule{2cm}{0.4pt}.
Step-by-Step Solution
Key Concept: Identify the sequence as an A.P.\ ($a=8,\,d=13$) and find $n$ from the last term. Use $\sigma^{2}=\dfrac{\sum x_{i}^{2}}{n}-\bar{x}^{2}$ with $\bar{x}=\dfrac{a+\ell}{2}$ for an A.P.
From $8+(n-1)\cdot 13=320$: $n=25.$ Mean $\bar{x}=\dfrac{8+320}{2}=164.$
$\displaystyle\sum_{i=1}^{25}x_{i}^{2}=\sum_{k=0}^{24}(8+13k)^{2}=25(64)+2\cdot 8\cdot 13\sum_{k=0}^{24}k+13^{2}\sum_{k=0}^{24}k^{2}$
$=1600+208\cdot 300+169\cdot\dfrac{24\cdot 25\cdot 49}{6}=1600+62400+828100=892100.$
$$\sigma^{2}=\frac{892100}{25}-164^{2}=35684-26896=8788.$$
Correct Answer: 8788