Question

The following table shows the ages of the patients admitted in a hospital during a year:

Age (in years):	$5-15$	$15-25$	$25-35$	$35-45$	$45-55$	$55-65$
No. of students:	$6$	$11$	$21$	$23$	$14$	$5$

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Answer 1

$\Rightarrow$ Now taking $30$ as assumed mean $\left(a\right)$ we may calculate $d_i$ and $f_i{d}_i$ as follows.

$Age$	$Numberofpatients$ $f_i$	$AverageMarks$ $x_i$	$d-i=x_i-30$	$f_i{d}_i$
$5-15$	$6$	$10$	$-20$	$-120$
$15-25$	$11$	$20$	$-10$	$-110$
$25-35$	$21$	$30$	$0$	$0$
$35-45$	$23$	$40$	$10$	$230$
$45-55$	$14$	$50$	$20$	$280$
$55-65$	$5$	$60$	$30$	$150$
$Total$	$80$			$430$

From the table we may observe that, $\sum f_i=80$ and $\sum f_i{d}_i=430$.

$Mean=a+\frac{\sum f_i{d}_i}{\sum f_i}$

$a=$ assumed mean $=30$

$\Rightarrow$ $Mean=30+\frac{430}{80}$

$\Rightarrow$ $Mean=30+5.375$

$\therefore$ $Mean=35.375\approx 35.38$

Clearly, mean of this data is 35.38. It represents that on an average the age of a patients admitted to hospital was $35.38years.$

$\Rightarrow$ As we may observe that maximum class frequency is 23 belonging to class interval $35–45$

$\Rightarrow$ So, modal class $=35–45$

We know,

$Mode=l+\frac{f-f_1}{2f-f_1-f_2}\times h$

$l=$ lower limit of the modal class

$h=$ size of the class intervals

$f=$ frequency of the modal class

$f_1=$ frequency of the class preceding the modal class

$f_2=$ frequency of the class succeed in the modal class.

$\Rightarrow$ $l=35,f=23,h=10,f_1=21$ and $f_2=14$.

$\Rightarrow$ $Mode=35+\frac{23-21}{2\times 23-21-14}\times 10$

$\Rightarrow$ $Mode=35+\frac{2}{46-35}\times 10$

$\Rightarrow$ $Mode=35+1.81=36.8$

$\therefore$ Clearly mode is $\mathrm{36.8.}$ It represents that the maximum number of patients admitted in the hospital was of $36.8years.$

The mode of the data shows that maximum number of patients is in the age group of 36.8, while average age of all the patients is 35.38.