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.