Calculate the mean and cumulative frequency for the given marks using a frequency table.

The highest frequency in the table is 12.

The class interval corresponding to the frequency 12 is $60-64$.

The modal class is $\boxed{\text{60-64}}$.

The mean ($\bar{x}$) is given by the formula:

From the table, $\sum fx = 2995$ and $\sum f = 50$.

The estimated mean mark is $\boxed{59.9}$.

The total number of students ($N$) is 50.
The position of the median is $\frac{N}{2} = \frac{50}{2} = 25^{th}$ value.

From the cumulative frequency column, the $25^{th}$ value falls within the class $60-64$ (since the cumulative frequency for $55-59$ is 23, and for $60-64$ it is 35).
So, the median class is $60-64$.

• Lower class boundary of the median class ($L$) = 59.5
• Total frequency ($N$) = 50
• Cumulative frequency of the class before the median class ($C_f$) = 23 (for class $55-59$)
• Frequency of the median class ($f_m$) = 12
• Class width ($c$) = $64.5 - 59.5 = 5$

The estimated median is $\boxed{60.33}$.