Adding to the mean

$\text{Given that, } \bar{x} = 19.3. \\ \implies \frac{\sum x_i}{\sum f_i} = 19.3 \\ \text{Here total no of observations, }\sum f_i \text{, is equal to 20} \\ \implies \frac{\sum x_i}{20} = 19.3 \\ \implies \sum x_i = 19.3 \times 20 \\ \implies \sum x_i = 386 \\ \text{Now, on adding a new observation, }y \text{ to the old sum of observations, }\sum x_i + y \text{ The new mean is decreased by 0.5} \\ \implies \frac{\sum x_i + y}{21} = 19.3 - 0.5 \text{ (21 no of observations, as previously you had 20 observations, on adding one new observation 21.)} \\ \implies \sum x_i + y = 394.8 \\ \implies 386 + y = 194.8 \\ \implies y = 8.8 \\ \text{Therefore, the new observation added is 8.8}$