Ex.14.2 Q1 STATISTICS Solution - NCERT Maths Class 10
Question
The following table shows the ages of the patients admitted in a hospital during a year:
Age ( in years) | \(5-15\) | \(15 - 25\) | \(25 - 35\) | \(35 - 45\) | \(45 - 55\) | \(55 - 65\) |
Number of Patients | \(6\) | \(11\) | \(21\) | \(23\) | \(14\) | \(5\) |
Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
Text Solution
What is known?
The ages of the patients admitted in a hospital during a year.
What is unknown?
The mode and the mean of the data and their comparison and 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,\({x_i} = \frac{{{\text{Upper class limit }} + {\text{ Lower class limit}}}}{2}\)
Age (in years) |
Number of patients \(f_i\) |
\(x_i\) |
\(f_i\, x_i\) |
5 – 15 |
6 |
10 |
6 |
15 – 25 |
11 |
20 |
220 |
25 – 35 |
21 |
30 |
630 |
35 – 45 |
23 |
40 |
920 |
45 – 55 |
14 |
50 |
700 |
55 – 65 |
5 |
60 |
300 |
\(\Sigma f_i = 80\) | \(\sum {{f_i}{x_i}} = 2830\) |
From the table it can be observed that,
\[\begin{array}{l}\sum {{f_i} = 80} \\\sum {{f_i}{x_i}} = 2830\end{array}\]
Mean,\(\overline x = \frac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\)
\(\begin{array}{l} = \frac{{2830}}{{80}}\\ = 35.37\end{array}\)
To find mode
We know that,Modal Class is the class with highest frequency
Age (in years) |
Number of patients \(f_i\) |
5 – 15 |
6 |
15 – 25 |
11 |
25 – 35 |
21 |
35 – 45 |
23 |
45 – 55 |
14 |
55 – 65 |
5 |
From the table, it can be observed that the maximum class frequency is \(23,\) belonging to class interval \(35 − 45.\)
Therefore, Modal class \(=35 − 45\)
Class size,\(h=10\)
Lower limit of modal class,\(l=35\)
Frequency of modal class,\(f_1\)
Frequency of class preceding modal class,\(f_0=23\)
Frequency of class succeeding the modal class,\(f_2=14\)
Mode,\( = l + \left( {\frac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h\)
\[\begin{align}&= 35 + \left( {\frac{{23 - 21}}{{2 \times 23 - 21 - 14}}} \right) \times 10\\ &= 35 + \left( {\frac{2}{{46 - 35}}} \right) \times 10\\ &= 35 + \frac{2}{{11}} \times 10\\&= 35 + 1.8\\&= 36.8\end{align}\]
So the modal age is \(36.82\) years which means maximum patients admitted to the hospital are of age \(36.82\) years .
Mean age is \(35.37\) and average age of the patients admitted is \(35.37\) years.