Let a _ 1 , a _ 2 , . . . , a _ 10 be 10 observations such that ∑ _...

Given: $\mathop{∑}_{k = 1}^{10}{a}_{k} = 50$

$⇒ {a}_{1} + {a}_{2} + \ldots + {a}_{10} = 50 . . . \left(i\right)$

$⇒ {\left({a}_{1} + {a}_{2} + \ldots + {a}_{10}\right)}^{2} = 2500$

$⇒ \mathop{∑}_{i = 1}^{10}{a}_{i}^{2} + 2\mathop{∑}_{k < j}{a}_{k}{a}_{j} = 2500$

$⇒ \mathop{∑}_{i = 1}^{10}{a}_{i}^{2} + 2 \times 1100 = 2500$

$⇒ \mathop{∑}_{i = 1}^{10}{a}_{i}^{2} = 2500 - 2\left(1100\right)$

$⇒ \mathop{∑}_{i = 1}^{10}{a}_{i}^{2} = 300$

 Standard deviation, $\sigma = \sqrt{\dfrac{∑ {a}_{i}^{2}}{n} - {\left(\dfrac{∑ {a}_{i}}{n}\right)}^{2}}$

$⇒ \sigma = \sqrt{\dfrac{∑ {a}_{i}^{2}}{10} - {\left(\dfrac{∑ {a}_{i}}{10}\right)}^{2}}$

$⇒ \sigma = \sqrt{\dfrac{300}{10} - {\left(\dfrac{50}{10}\right)}^{2}}$

$⇒ \sigma = \sqrt{30 - 25} = \sqrt{5}$