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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