Solution:

The frequency distribution is given

We now need to find the mean and standard deviation

Let’s construct a table of the given data and append other columns after calculations

$$\begin{tabular}{|l|l|l|}
\hline Marks $\left(\mathrm{x}_{i}\right)$ & Frequency $\left(f_{i}\right)$ & $f_{i} x_{i}$ \\
\hline 2 & 1 & 2 \\
\hline 3 & 6 & 18 \\
\hline 4 & 6 & 24 \\
\hline 5 & 6 & 30 \\
\hline 6 & 8 & 48 \\
\hline 7 & 2 & 14 \\
\hline 8 & 2 & 16 \\
\hline 9 & 3 & 27 \\
\hline 10 & 0 & 0 \\
\hline 11 & 2 & 22 \\
\hline 12 & 1 & 12 \\
\hline 13 & 0 & 0 \\
\hline 14 & 0 & 0 \\
\hline 15 & 0 & 0 \\
\hline 16 & 1 & 16 \\
\hline Total & $N=38$ & $\sum f_{i} x_{i}=229$ \\
\hline Here mean, & $\overline{\mathrm{x}}=\frac{\Sigma \mathrm{f}_{1} \mathrm{x}_{i}}{\mathrm{~N}}=\frac{229}{40}=6.02=6$ \\
\hline
\end{tabular}$$

Therefore the above table with more columns is shown below,

$$\begin{tabular}{|l|l|l|l|l|l|}
\hline Marks $\left(\mathrm{x}_{i}\right)$ & Frequency $\left(\mathrm{f}_{i}\right)$ & $\mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}$ & $d_{i}=x_{i}-\bar{x}$ & $\mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}$ & $\mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}^{2}$ \\
\hline 2 & 1 & 2 & $2-6=-4$ & $1 \times-4=-4$ & $1 \times-4^{2}=16$ \\
\hline 3 & 6 & 18 & $3-6=-3$ & $6 x-3=-18$ & $6 x-3^{2}=54$ \\
\hline 4 & 6 & 24 & $4-6=-2$ & $6 x-2=-12$ & $6 \times-2^{2}=24$ \\
\hline 5 & 6 & 30 & $5-6=-1$ & $6 \times-1=-6$ & $6 \times-1^{2}=6$ \\
\hline 6 & 8 & 48 & $6-6=0$ & $8 \times 0=0$ & $8 \times 0^{2}=0$ \\
\hline 7 & 2 & 14 & $7-6=1$ & $2 \times 1=2$ & $2 \times 1^{2}=2$ \\
\hline 8 & 2 & 16 & $8-6=2$ & $2 \times 2=4$ & $2 \times 2^{2}=8$ \\
\hline 9 & 3 & 27 & $9-6=3$ & $3 \times 3=9$ & $3 \times 3^{2}=27$ \\
\hline 10 & 0 & 0 & $10-6=4$ & $0 \times 4=0$ & $0 \times 4^{2}=0$ \\
\hline 11 & 2 & 22 & $11-6=5$ & $2 \times 5=10$ & $2 \times 5^{2}=50$ \\
\hline 12 & 1 & 12 & $12-6=6$ & $1 \times 6=6$ & $1 \times 6^{2}=36$ \\
\hline 13 & 0 & 0 & $13-6=7$ & $0 \times 7=0$ & $0 \times 7^{2}=0$ \\
\hline 14 & 0 & 0 & $14-6=8$ & $0 \times 8=0$ & $0 \times 8^{2}=0$ \\
\hline 15 & 0 & 0 & $15-6=9$ & $0 \times 9=0$ & $0 \times 9^{2}=0$ \\
\hline 16 & 1 & 16 & $16-6=10$ & $1 \times 10=10$ & $1 \times 10^{2}=100$ \\
\hline Total & $\mathrm{N}=38$ & $\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}$ $=229$ & & $\sum \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}=1$ & $\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}^{2}=323$ \\
\hline
\end{tabular}$$

And the standard deviation is

$\sigma=\sqrt{\frac{\sum \mathrm{f}_{i} \mathrm{~d}_{\mathrm{i}}^{2}}{\mathrm{n}}-\left(\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{d}_{i}}{\mathrm{n}}\right)^{2}}$

Substitute the values from the above table,

$\sigma=\sqrt{\frac{323}{38}-\left(\frac{1}{38}\right)^{2}}$

$\sigma=\sqrt{8.5-(0.026)^{2}}$

$\sigma=\sqrt{8.5-0.000676}=\sqrt{8.5}$

$\Rightarrow \sigma=2.9$

As a result, the mean and the standard deviation of the marks are respectively $6$ and $2.9$.