Ex.14.2 Q1 STATISTICS Solution - NCERT Maths Class 10

Go back to  'Ex.14.2'

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

\[\begin{array}{l}
 = 35 + \left( {\frac{{23 - 21}}{{2 \times 23 - 21 - 14}}} \right) \times 10\\
 = 35 + \left( {\frac{2}{{46 - 35}}} \right) \times 10\\
 = 35 + \frac{2}{{11}} \times 10\\
 = 35 + 1.8\\
 = 36.8
\end{array}\]

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.

  
Learn math from the experts and clarify doubts instantly

  • Instant doubt clearing (live one on one)
  • Learn from India’s best math teachers
  • Completely personalized curriculum

Frequently Asked Questions



What are Class 10 NCERT Exemplars?
While getting good scores in school tests is a desirable outcome, it is not a reliable indicator of how strong your child’s math foundation really is. Many students who score well in school exams in their earlier years, might struggle with math in higher grades because of a weak foundation. At Cuemath, we evaluate your child’s grasp of math fundamentals, and take corrective actions immediately. Also, your child may have limited exposure in their school, and in most cases, may not feel challenged to learn more. Cuemath's customised learning plan ensures your child is challenged with varied difficulty levels of questions at every stage.
What is the difference between CBSE and NCERT syllabus for Class 10?
How will Class 10 NCERT books help in exam preparation?
How will Class 10 NCERT books help you understand basic math concepts?
Which is the best video solution for the class 10 maths NCERT?