Ex.14.2 Q4 Statistics Solution - NCERT Maths Class 10
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\( = l + \left( {\frac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h\)
\[\begin{align} &= 30 + \left( {\frac{{10 - 9}}{{2 \times 10 - 9 - 3}}} \right) \times 5\\&= 30 + \left( {\frac{1}{{20 - 12}}} \right) \times 5\\&= 30 + \frac{5}{8}\\&= 30 + 0.625\\&= 30.625\\&= 30.6\end{align}\]
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 \(29.26\)