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Question

# Let ${x}_{1},{x}_{2},...,{x}_{n}$ be n observations and $\overline{)X}$ be their arithmetic mean. The standard deviation is given by (a) $\sum _{i=1}^{n}{\left({x}_{i}-\overline{)X}\right)}^{2}$ (b) $\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{)X}\right)}^{2}$ (c) $\sqrt{\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{)X}\right)}^{2}}$ (d) $\sqrt{\frac{1}{n}\sum _{i=1}^{n}{x}_{i}^{2}-{\overline{)X}}^{2}}$

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Solution

## It is given that ${x}_{1},{x}_{2},...,{x}_{n}$ are n observations and $\overline{)X}$ is their arithmetic mean. The standard deviation of given observations is $\sqrt{\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{)X}\right)}^{2}}$. Also, $\sqrt{\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{)X}\right)}^{2}}$ = $\sqrt{\frac{1}{n}\sum _{i=1}^{n}{x}_{i}^{2}-{\overline{)X}}^{2}}$ Hence, the correct answers are options (c) and (d). Disclaimer: For option (c) to be the only correct answer, option (d) should be different from the given value.

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