If the mean and variance of five observations are (24)/(5) and...

Given: 

$x = \dfrac{24}{5}$ and variance is $\dfrac{194}{25}$

$⇒ \mathop{∑}_{i = 1}^{5}{x}_{i} = \dfrac{24}{5} \times 5$

$⇒ \mathop{∑}_{i = 1}^{5}{x}_{i} = 24$

Also, variance

$⇒ \dfrac{\mathop{∑}_{i = 1}^{5}{\left({x}_{i}\right)}^{2}}{5} - {\left(\dfrac{24}{5}\right)}^{2} = \dfrac{194}{25}$

$⇒ \mathop{∑}_{i = 1}^{5}{\left({x}_{i}\right)}^{2} = 154$

Now, ${5}^{th}$ observation $= 24 - \text{sum of first}4\text{terms}$

$= 24 - \dfrac{7}{2} \times 4 = 10$

So, new variance $= \dfrac{\mathop{∑}_{i = 1}^{4}{\left({x}_{i}\right)}^{2}}{4} - {\left(\dfrac{7}{2}\right)}^{2}$

$= \dfrac{154 - 100}{4} - \dfrac{49}{4}$

$= \dfrac{5}{4}$