Question

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

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

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

Answer 1

Since the maximum frequency is $23$ and it corresponds to the class $35-45$.
$\therefore$ Modal class $=35-45$

We know that,

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

Where, $l\to$ lower limit of the modal class

$f\to$ frequency of the modal class

$f_1\to$ frequency of the class preceding the modal class

$f_2\to$ frequency of the class succeeding the modal class

$h\to$ class width

Here, $l=35,h=10,f=23,f_1=21,f_2=14$
$\therefore$ Mode $=35+\left(\frac{23-21}{2\left(23\right)-21-14}\right)\times 10$

$=35+\frac{10\left(2\right)}{11}$
$=35+\frac{20}{11}$

$=35+1.82$

$=36.82$ years (approximately)

Again, we have:

Age (in years)	Number of patients $\left(f_i\right)$	Mid-value $\left(x_i\right)$	$f_i{x}_i$
$5-15$	$6$	$10$	$60$
$15-25$	$11$	$20$	$220$
$25-35$	$21$	$30$	$630$
$35-45$	$23$	$40$	$920$
$45-55$	$14$	$50$	$700$
$55-65$	$5$	$60$	$300$
	$\sum f_i=80$		$\sum f_i{x}_i=2830$

We know that, Mean, $\overline{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

$\therefore \overline{x}=\frac{2830}{80}$

$=\frac{283}{8}$

$=35.38$ years

Hence, for the given data, mean $=35.38$ years and mode $=36.82$ years.

This shows that maximum number of patients admitted in the hospital are of the age $36.82$ years, while on an average the age of a patient admitted to the hospital is $35.38$ years.