Ans: (a) The mean $=5.4$, the median $=5.5$, the standard deviation $=0.917$ (b) The median decreases by $0.5$.
- The mean
$\begin{array}{cl}
= & \dfrac{4 \times 8 + 5 \times 12 + 6 \times 16 + 7 \times 4}{8 + 12 + 16 + 4} \\
= & 5.4
\end{array}$Since there are $40$ students, then the median
$\begin{array}{cl}
= & \dfrac{\text{the 20th datum} + \text{the 21st datum}}{2} \\
= & \dfrac{5 + 6}{2} \\
= & 5.5
\end{array}$The standard deviation
$\begin{array}{cl}
= & 0.976\ 515\ 139
\end{array}$ - After adding a new student, the number of students becomes $41$. Then the new median
$\begin{array}{cl}
= & \text{the 21st datum} \\
= & 5
\end{array}$Therefore, the decrease in the median
$\begin{array}{cl}
= & 5.5 – 5 \\
= & 0.5
\end{array}$
