The marks distribution of $30$ students in a mathematics examination is given in the table. Find the mode of this data. Also,compare and interpret the mode and the mean.

Class interval	$10-25$	$25-40$	$40-55$	$55-70$	$70-85$	$85-100$
Number of students	$2$	$3$	$7$	$6$	$6$	$6$

(52) To find the mean,we calculate the class marks $(x_i)$ and $f_i x_i$:

Class interval	Number of students $(f_i)$	Class mark $(x_i)$	$f_i x_i$
$10-25$	$2$	$17.5$	$35.0$
$25-40$	$3$	$32.5$	$97.5$
$40-55$	$7$	$47.5$	$332.5$
$55-70$	$6$	$62.5$	$375.0$
$70-85$	$6$	$77.5$	$465.0$
$85-100$	$6$	$92.5$	$555.0$
Total	$\Sigma f_i = 30$		$\Sigma f_i x_i = 1860.0$

Mean $\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{1860}{30} = 62$.
For the mode,the maximum frequency is $7$,which corresponds to the modal class $40-55$.
Here,$l = 40$,$h = 15$,$f_1 = 7$,$f_0 = 3$,$f_2 = 6$.
Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 40 + \left( \frac{7 - 3}{14 - 3 - 6} \right) \times 15 = 40 + \left( \frac{4}{5} \right) \times 15 = 40 + 12 = 52$.
Interpretation: The maximum number of students obtained $52$ marks (mode),while on average,a student obtained $62$ marks (mean).