Let  $x_1,x_2,\mathrm{......},x_{n}be$  n observations such that  $\sum _{ }^{ }{X}_i^2$=400 and $\sum _{ }^{ }{X}_i^{ }$=80. Then   the least possible value of n  is

we know that for positive real numbers  $x_1,x_{2,\dots ,\dots }{x}_n$

A.M   of k^th  power of  $x_i\ge k^{th}$  the power of A.M of x_i
$\Rightarrow \sum _n x_1^2\ge {\left(\frac{\sum x_1}{n}\right)}^2\Rightarrow \frac{400}{n}\ge {\left(\frac{80}{n}\right)}^2\Rightarrow n\ge 16$
so least possible value for n=16