Question

The mean of $30$ items is $18$ and their standard deviation is $3$. Find the sum of all the items and also the sum of the squares of all the items.

Answer 1

The mean of $30$ items, $\overline{x}=18$
The sum of $30$ items, $\sum x=30\times 18=540$ $(\overline{x}=\frac{\sum x}{n})$
Standard deviation, $\sigma =3$
Now, ${\sigma }^2=\frac{\sum x^2}{n}-{\left(\frac{\sum x}{n}\right)}^2$
$\Rightarrow \frac{\sum x^2}{30}-{18}^2=9$
$\Rightarrow \frac{\sum x^2}{30}-324=9$
$\Rightarrow \sum x^2-9720=270$
$\sum x^2=9990$
$\therefore \sum x=540$ and $\sum x^2=9990$.