Statistics
Standard Deviation from Sum and Sum of Products
nta_pyq_2024_jan
Grade 11
Question:
Let $a_1,a_2,\ldots,a_{10}$ be 10 observations such that $\displaystyle\sum_{k=1}^{10}a_k=50$ and $\displaystyle\sum_{\forall k<j}a_k\cdot a_j=1100$. Then the standard deviation of $a_1,a_2,\ldots,a_{10}$ is equal to:
5
$\sqrt{5}$
10
$\sqrt{115}$
Step-by-Step Solution
Key Concept: $\left(\sum a_k\right)^2=\sum a_k^2+2\sum_{k<j}a_ka_j=2500$. $\sum a_k^2=2500-2(1100)=300$. $\sigma=\sqrt{\frac{\sum a_k^2}{10}-\left(\frac{\sum a_k}{10}\right)^2}=\sqrt{30-25}=\sqrt5$.
$\sigma=\sqrt{30-25}=\sqrt{5}$.
Correct Answer: 2