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.

(N/A) To find the class marks $(x_{i})$,the following relation is used:
$x_{i} = \frac{\text{Upper class limit} + \text{Lower class limit}}{2}$
Taking $30$ as the assumed mean $(a)$,$d_i$ and $f_id_i$ are calculated as follows:

Age (in years)	Number of patients $(f_i)$	Class mark $(x_i)$	$d_i = x_i - 30$	$f_i d_i$
$5$-$15$	$6$	$10$	-$20$	-$120$
$15$-$25$	$11$	$20$	-$10$	-$110$
$25$-$35$	$21$	$30$	$0$	$0$
$35$-$45$	$23$	$40$	$10$	$230$
$45$-$55$	$14$	$50$	$20$	$280$
$55$-$65$	$5$	$60$	$30$	$150$
Total	$80$	-	-	$430$

From the table,we obtain $\Sigma f_{i} = 80$ and $\Sigma f_{i} d_{i} = 430$.
Mean,$\bar{x} = a + \frac{\Sigma f_{i} d_{i}}{\Sigma f_{i}} = 30 + \frac{430}{80} = 30 + 5.375 = 35.375 \simeq 35.38$.
The mean age of the patients is $35.38 \text{ years}$. This represents that,on average,the age of a patient admitted to the hospital was $35.38 \text{ years}$.
It can be observed that the maximum class frequency is $23$,belonging to the class interval $35-45$.
Modal class $= 35-45$,Lower limit $(l) = 35$,Class size $(h) = 10$,Frequency $(f_1) = 23$,Frequency $(f_0) = 21$,Frequency $(f_2) = 14$.
Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 35 + \left( \frac{23 - 21}{2(23) - 21 - 14} \right) \times 10 = 35 + \left( \frac{2}{46 - 35} \right) \times 10 = 35 + \frac{20}{11} = 35 + 1.81 = 36.81$.
The mode is $36.81$. This represents that the age of the maximum number of patients admitted to the hospital was $36.81 \text{ years}$.