Question

The following tables gives 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

• A. mode $=38.8$, mean $=33.37$
• B. Mode $=36.8$, Mean $=35.37$
• C. Mode $=35.8$, Mean$=36.37$
• D. Mode $=38.6$, Mean$=37.35$

Answer 1

The correct option is B Mode $=36.8$, Mean $=35.37$

Consider the following table to calculate Mode & Mean:

$ci$	$f_i$	$x_i$	$f_i{x}_i$
$5-15$	$6$	$10$	$60$
$15-25$	$11$	$20$	$220$
$25-35$	$21$	$30$	$630$
$35-45$	$23$	$40$	$920$
$45-55$	$14$	$50$	$700$
$55-65$	$5$	$60$	$300$

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

Here,

maximum frequency $f_1=23$

$\Rightarrow$ modal class $=35-45$

lower limit of modal class $l=35$

class size $h=10$

preceding frequency of modal class $f_0=21$

succeeding frequency of modal class $f_2=14$

$\therefore Mode=35+(\frac{23-21}{46-21-14})\times 10$

$\therefore Mode=35+(\frac{2}{11})\times 10$

$\therefore Mode=36.81\approx 36.8$

$N=\Sigma f_i=80$

$\Sigma f_i{x}_i=2830$

Mean $\overline{x}=\frac{\Sigma f_i{x}_i}{N}$

$\therefore \overline{x}=\frac{2830}{80}=35.37$

Hence, option $B$ is correct.