Find the mean and the mode of the following frequency distribution :
Class Frequency
$\displaystyle 0$-$\displaystyle 15$ $\displaystyle 9$
$\displaystyle 15$-$\displaystyle 30$ $\displaystyle 15$
$\displaystyle 30$-$\displaystyle 45$ $\displaystyle 35$
$\displaystyle 45$-$\displaystyle 60$ $\displaystyle 20$
$\displaystyle 60$-$\displaystyle 75$ $\displaystyle 11$
$\displaystyle 75$-$\displaystyle 90$ $\displaystyle 13$
$\displaystyle 90$-$\displaystyle 105$ $\displaystyle 17$

Class	\(\displaystyle f_{i}\)	\(\displaystyle \boldsymbol{x}_{\boldsymbol{i}}\)	\(\displaystyle \boldsymbol{u}_{\boldsymbol{i}}=\frac{\boldsymbol{x}_{\boldsymbol{i}}-\mathbf{5 2 . 5}}{\mathbf{1 5}}\)	\(\displaystyle f_{i} u_{i}\)
$\displaystyle 0$-$\displaystyle 15$	$\displaystyle 9$	$\displaystyle 7.5$	-$\displaystyle 3$	-$\displaystyle 27$
$\displaystyle 15$-$\displaystyle 30$	$\displaystyle 15$	$\displaystyle 22.5$	-$\displaystyle 2$	-$\displaystyle 30$
$\displaystyle 30$-$\displaystyle 45$	$\displaystyle 35$	$\displaystyle 37.5$	-$\displaystyle 1$	-$\displaystyle 35$
$\displaystyle 45$-$\displaystyle 60$	$\displaystyle 20$	$\displaystyle 52.5$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle 60$-$\displaystyle 75$	$\displaystyle 11$	$\displaystyle 67.5$	$\displaystyle 1$	$\displaystyle 11$
$\displaystyle 75$-$\displaystyle 90$	$\displaystyle 13$	$\displaystyle 82.5$	$\displaystyle 2$	$\displaystyle 26$
$\displaystyle 90$-$\displaystyle 105$	$\displaystyle 17$	$\displaystyle 97.5$	$\displaystyle 3$	$\displaystyle 51$
Total	$\displaystyle 120$			-$\displaystyle 4$

Marking-scheme solution