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.

Answer 1

Mode: The class which have highest frequency.

In this case, class interval $35-45$ is the modal class.

Now,

Lower limit of modal class, $l=35,h=10,f_1,=23,f_0=21,f_2=14$

We know that,

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

$=35+\left(\frac{23-21}{2\left(23\right)-21-14}\right)\times 10$

$=35+\frac{2}{11}\times 10$

$=35+1.818$

Mode $=36.8$

Lets take assumed mean, $A$ as $30$

Class interval $=10$

$\therefore u_i=\frac{x_i-A}{h}=\frac{x_i-30}{10}$

$\overline{x}=A+h\frac{\sum f_i{u}_i}{\sum f_i}=30+10\times \frac{43}{80}$

$=30+5.375$
$=35.375$

$\approx 35.37$

Hence, Mode $=36.8$ years, Mean $=35.37$ years.

Maximum number of patients admitted in the hospital are of age group $36.8$ years, while on an average the age of a patient admitted to the hospital is $35.37$ years.