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
15 - 20	3
20 - 25	8
25 -30	9
30 - 35	10
35 - 40	3
40 - 45	0
45 - 50	0
50 - 55	2

Answer 1

From the data given above, it can be observed that the maximum class frequency is 10, belonging to class interval 30 -35.

Therefore, modal class = 30 -35
Lower limit ($l$) of modal class = 30
Frequency ($f_1$) of modal class = 10
Frequency ($f_0$) of class preceding the modal class = 9
Frequency ($f_2$) of class succeeding the modal class = 3
Class size ($h$) = 5

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

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

Mode =$30+ [\frac{1}{20 - 12}] \times 5$

Mode = $30 +( \frac{5}{ 8})$
Mode = 30 + 0.625
Mode = 30.6

It represents that most of the states/U.T have a teacher-student ratio as 30.6.

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To find the class mark ($x_i$) for each interval, the following relation is used.

Class mark $(x_i)$ = $\frac {\text{Upper \,limit + Lower \,limit}}{2}$

Taking 32.5 as assured mean (a), $d_i$, $u_i$, and $f_iu_i$ can be calculated as follows.

**Number of students per teacher **	**Number of states/U.T (fi) **	** $\bf{x_i}$ **	$\bf{d_i = x_i -32.5}$	$\bf{u_i = \frac{d_i}{5}}$	$\bf{f_iu_i}$
15 - 20	3	17.5	-15	-3	-9
20 - 25	8	22.5	-10	-2	-16
25 - 30	9	27.5 -5	-5	-1	-9
30 - 35	10	32.5	0	0	0
35 - 40	3	37.5	5	1	3
40 - 45	0	42.5	10	2	0
45 - 50	0	47.5	15	3	0
50 - 55	2	52.5	20	4	8
**Total **	35				-23

From the table, it can be observed that

Mean, $\overset{-}{x} = a + (\frac{\sum f_iu_i}{\sum f_i})h$

x = $32.5 + (\frac{-23 }{35})\times 5$

x = $32.5 -\frac{23}7$
x = 32.5 - 3.28
x = 29.22

Therefore, mean of the data is 29.2.
It represents that on an average, teacher−student ratio was 29.2.