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) ${\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{X}\right)}^2$ (b) $\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{X}\right)}^2$ (c) $\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{X}\right)}^2}$ (d) $\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2-{\overline{X}}^2}$

Answer 1

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.