Question

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret, the two measures.

Number of students per teacher	Number of States/U.T	Number of students per Teacher	Number of States.U.T
15-20	$3$	35-40	$3$
20-25	$8$	40-45	$0$
25-30	$9$	45-50	$0$
30-35	$10$	50-55	$2$

Answer 1

No. of students per teacher	$x_i$	No. of states/U.T. $\left(f_i\right)$	$f_i{x}_i$
15-20	17.5	3	52.5
20-25	22.5	8	180
25-30	27.5	9	247.5
30-35	32.5	10	325
35-40	37.5	3	112.5
40-45	42.5	0	0
45-50	47.5	0	0
50-55	52.5	2	105
		$\Sigma f_i=35$	$\Sigma f_i{x}_i=1022.5$

$Mean=\frac{\Sigma f_i{x}_i}{\Sigma f_i}=\frac{1022.5}{35}=29.21$

As frequency is maximum for the class 30-35, modal class is 30-35.

$Mode=l+\frac{f_1-f_0}{2f_1-(f_0+f_2)}\times h$

where,

$l=$ lower limit of modal class = 30

$f_1$ = frequency of modal class = 10

$f_0$ = frequency of class preceding the modal class = 9

$f_2$ = frequency of class succeding the modal class = 3

$h=$ higher limit - lower limit = 35 - 30 = 5

$Mode=30+\frac{10-9}{2\times 10-(9+3)}\times 5$

$\Rightarrow mode=30+\frac{1}{8}\times 5=30+0.625=30.625$

Hence, the mean and modal given data are $29.21$ and $30.625$ respectively.