Ex.14.2 Q4 Statistics Solution - NCERT Maths Class 10

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Question

The following data gives the state- wise teacher- student ratio in higher secondary schools of India. Find the mode and mean of the data and interpret the two.

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

Text Solution

What is known?

The state- wise teacher- student ratio in higher secondary schools of India.

What is unknown?

The mode and mean of the data and their interpretation.

Reasoning:

We will find the mean by direct method.

Mean,$$\overline x = \frac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$$

Modal Class is the class with highest frequency

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

Where,

Class size,$$h$$

Lower limit of modal class,$$l$$

Frequency of modal class,$$f_1$$

Frequency of class preceding modal class,$$f_0$$

Frequency of class succeeding the modal class,$$f_2$$

Steps:

To find mean

We know that,

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

Class size,$$h=500$$

Taking assumed mean,$$a=2750$$

 Number of students per teacher Number of states / U.T. $$f_i$$ ${x_i}$ ${f_i}{x_i}$ 15 – 20 3 17.5 52.5 20 – 25 8 22.5 180 25 – 30 9 27.5 247.5 30 – 35 10 32.5 325 35 – 40 3 37.5 112.5 40 – 45 0 42.5 0 45 – 50 0 47.5 0 50 – 55 2 52.5 105 $\sum {{f_i}} = 35$ $\sum {{f_i}{x_i}} = 1024$

Mean, $$\overline x = \frac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$$

$\begin{array}{l} = \frac{{1024}}{{35}}\\ = 29.26 \end{array}$

To find mode

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

From the table, it can be observed that the maximum class frequency is $$10,$$ belonging to class interval $$30 − 35$$

Therefore, Modal class$$=30 − 35$$

Class size,$$h=5$$

Lower limit of modal class,$$l=30$$

Frequency of modal class,$$f_1=10$$

Frequency of class preceding modal class,$$f_0=9$$

Frequency of class succeeding the modal class,$$f_2=3$$

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

$\begin{array}{l} = 30 + \left( {\frac{{10 - 9}}{{2 \times 10 - 9 - 3}}} \right) \times 5\\ = 30 + \left( {\frac{1}{{20 - 12}}} \right) \times 5\\ = 30 + \frac{5}{8}\\ = 30 + 0.625\\ = 30.625\\ = 30.6 \end{array}$

The modal teacher- student ratio is $$30.6$$ and mean teacher- student ratio is $$29.26.$$

Most states/U.T. have a teacher- student ratio of $$30.6$$ and on an average the ratio is $$29.26$$

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