Browse Questions

# The standard deviation of $n$ observations $x_1, x_2, ....x_n$ is 2. If $\sum_{i=1}^n x_i=20$ and $\sum_{i=1}^n x_i^2=100$ then $n$ is $i=1$

$\begin {array} {1 1} (A)\;10 \: or \: 20 & \quad (B)\;5 \: or \: 10 \\ (C)\;5 \: or \: 20 & \quad (D)\;5 \: or \: 15 \end {array}$

We know SD = $\sqrt{ \large\frac{\Sigma x_i^2}{n} - \bigg( \large\frac{ \Sigma x_i}{n} \bigg)^2}$
$\therefore 2 = \sqrt{ \large\frac{100}{n} - \bigg( \large\frac{20}{n} \bigg)^2}$
$=\sqrt{ \large\frac{100}{n}-\large\frac{400}{n^2}}$
$\Rightarrow 4 = \large\frac{100}{n}$ $- \large\frac{400}{n^2}$
$n^2-25n+100=0$
$n = 20, 5$
Ans : (C)