Question

Question 4
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.

$\begin{array}{cc}\text{Number of students per teacher}& \text{Number of states/U.T}\\ 15-20& 3\\ 20-25& 8\\ 25-30& 9\\ 30-35& 10\\ 35-40& 3\\ 40-45& 0\\ 45-50& 0\\ 50-55& 2\end{array}$

Answer 1

We may observe from the given data that maximum class frequency is 10 which belongs to class interval 30 - 35.

So, modal class = 30 - 35

Class size $h$ = 5

Lower limit $l$ of modal class = 30

Frequency $\left(f_1\right)$ of modal class = 10

Frequency $\left(f_0\right)$ of class preceding modal class = 9

Frequency $\left(f_2\right)$ of class succeeding modal class = 3

$Mode=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h=30+\left(\frac{10-9}{2\left(10\right)-9-3}\right)\times \left(5\right)=30+\left(\frac{1}{20-12}\right)5=30+\frac{5}{8}=30.625$

Mode = 30.6

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

Now, we may find the class marks by using the relation:

$Classmark=\frac{Upperclasslimit+lowerclasslimit}{2}$

Now, taking 32.5 as assumed mean $a$, we may calculate $d_i,u_{i}and{f}_i{u}_i$ as following:

$\begin{array}{cccccc}Numberofstudentsperteacher& Numberofstates/U.T\left(f_i\right)& x_i& d_i=x_i-32.5& u_i=\frac{x_i-32.5}{h}& f_i{u}_i\\ 15-20& 3& 17.5& -15& -3& -9\\ 20-25& 8& 22.5& -10& -2& -16\\ 25-30& 9& 27.5& -5& -1& -9\\ 30-35& 10& 32.5& 0& 0& 0\\ 35-40& 3& 37.5& 5& 1& 3\\ 40-45& 0& 42.5& 10& 2& 0\\ 45-50& 0& 47.5& 15& 3& 0\\ 50-55& 2& 52.5& 20& 4& 8\\ Total& 35& & & & -23\end{array}$

$Nowmean\overline{x}=a+\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)h32.5+\left(\frac{-23}{35}\right)\times 5=32.5-\frac{23}{7}=32.5-3.28$

= 29.22

So, mean of data is 29.2.

It represents that on an average teacher-student ratio is 29.2.