Question

If the mean and S.D. of n observation $x_1,x_2,.......x_n$ are $\overline{x}$ and $\sigma$ resp, then the sum of squares of observations is

• A. $n({\sigma }^2+{\overline{x}}^2)$
• B. $n({\sigma }^2-{\overline{x}}^2)$
• C. $n({\overline{x}}^2-{\sigma }^2)$
• D. none of these

Answer 1

The correct option is A $n({\sigma }^2+{\overline{x}}^2)$

$\because \sigma =\sqrt{\frac{\sum x_i^2}{n}-{\left(\frac{\sum x_i}{n}\right)}^2}$
$\sigma =\sqrt{\frac{\sum x_i^2}{n}-{\overline{x}}^2}$
Squaring both sides , we get
${\sigma }^2=\frac{\sum x_i^2}{n}-{\overline{x}}^2$
$\Rightarrow \sum x_i^2=n({\sigma }^2+{\overline{x}}^2)$