Mathematics - Statistics Question with Solution | TestHub

Let $a, b, c, d, e$ be $5$ unknown observations.

Given that No. of observation, $n=7,$ Mean $=8,$ Variance $=16$

$\therefore$ sum of observations  $=\mathrm{No}. \mathrm{of} \mathrm{observation} \times \mathrm{Mean}=7\times 8=56$

$\Rightarrow$ Mean of $5$ remaining observations $=\frac{56-8-6}{5}=\frac{42}{5}$

As, we know that,

Variance for $7$ observations$=\frac{\Sigma x_i^2}{n}-{\left(\overline{x}\right)}^2$

$\Rightarrow 16=\frac{\Sigma x_i^2}{7}-64$

$\Rightarrow \Sigma x_i^2=560$

$\Rightarrow a^2+b^2+c^2+d^2+e^2+64+36=560$

$\Rightarrow a^2+b^2+c^2+d^2+e^2=460$

Variance of remaining $5$ observations $=\frac{\Sigma x_i^2}{n}-{\left(\overline{x}\right)}^2=\frac{460}{5}-{\left(\frac{42}{5}\right)}^2=\frac{536}{25}$