The mean and variance of 7 observations are 8 and 16 respetively....

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

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

$∴$ sum of observations $= No . ofobservation \times Mean = 7 \times 8 = 56$

$⇒$ Mean of $5$ remaining observations $= \dfrac{56 - 8 - 6}{5} = \dfrac{42}{5}$

As, we know that,

Variance for $7$ observations$= \dfrac{Σ{x}_{i}^{2}}{n} - {\left(\overset{¯}{x}\right)}^{2}$

$⇒ 16 = \dfrac{Σ{x}_{i}^{2}}{7} - 64$

$⇒ Σ{x}_{i}^{2} = 560$

$⇒ {a}^{2} + {b}^{2} + {c}^{2} + {d}^{2} + {e}^{2} + 64 + 36 = 560$

$⇒ {a}^{2} + {b}^{2} + {c}^{2} + {d}^{2} + {e}^{2} = 460$

Variance of remaining $5$ observations $= \dfrac{Σ{x}_{i}^{2}}{n} - {\left(\overset{¯}{x}\right)}^{2} = \dfrac{460}{5} - {\left(\dfrac{42}{5}\right)}^{2} = \dfrac{536}{25}$