# Ex.14.2 Q1 STATISTICS Solution - NCERT Maths Class 10

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## 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.

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