The mean and variance of the marks obtained by the students in a...

Let the number of observations of the data set is $n$.

So,

$\mathop{∑}_{}{x}_{i} = 10n . . . . \left(i\right)$

Also,

$\mathop{∑}_{}{x}_{i} - 8 + 12 = 10 . 2n$

$⇒ 10n + 4 = 10 . 2n$

$⇒ 0 . 2n = 4$

$⇒ n = 20$

Also,

${\sigma}^{2} = 4 = \dfrac{\mathop{∑}_{}{x}_{i}^{2}}{20} - 100$

$⇒ \mathop{∑}_{}{x}_{i}^{2} = 104 \times 20 = 2080$

Now, new variance is

$= \dfrac{\mathop{∑}_{}{x}_{i}^{2} - {8}^{2} + {12}^{2}}{20} - {\left(10 . 2\right)}^{2}$

$= \dfrac{2080 + 4 \times 20}{20} - 104 . 04$

$= 108 - 104 . 04$

$= 3 . 96$