Question

The following table shows the ages of the patients admitted in a hospital during a year: Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Age (in $\mathrm{years}$)	$5-15$	$15-25$	$25-35$	$35-45$	$45-55$	$55-65$
Number of patients	$6$	$11$	$21$	$23$	$14$	$5$

Answer 1

Step 1: Determination of mode

Class Interval	Frequency (${\mathrm{f}}_{\mathrm{i}}$)	Mid-point (${\mathrm{x}}_{\mathrm{i}}$)	${\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}$
$5-15$	$6$	$10$	$60$
$15-25$	$11$	$20$	$220$
$25-35$	$21$$=\left({\mathrm{f}}_1\right)$	$30$	$630$
$35-45$	$23$ $=\left({\mathrm{f}}_{\mathrm{m}}\right)$	$40$	$920$
$45-55$	$14=\left({\mathrm{f}}_2\right)$	$50$	$700$
$55-65$	$5$	$60$	$300$
	$\sum {\mathrm{f}}_{\mathrm{i}}=80$		$\sum {\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}=2830$

The greatest frequency in the given table is $23$, so the modal class $=35-45,$

lower limit $\left(\mathrm{l}\right)=35,$Class width $=10$, ${\mathrm{f}}_{\mathrm{m}}=23,{\mathrm{f}}_1=21$ and ${\mathrm{f}}_2=14$

$\mathrm{Mode}=\mathrm{l}+\left(\frac{{\mathrm{f}}_{\mathrm{m}}-{\mathrm{f}}_1}{2{\mathrm{f}}_{\mathrm{m}}-{\mathrm{f}}_1-{\mathrm{f}}_2}\right)\times \mathrm{h}$

On substituting the values in the formula, we get

$\begin{array}{rcl}\mathrm{Mode}& =& 35+\left(\frac{23-21}{46-21-14}\right)\times 10\\ & =& 35+\frac{20}{11}\\ & =& 35+1.8\\ & =& 36.8\mathrm{years}\end{array}$

$\Rightarrow \mathrm{Mode}=36.8\mathrm{year}$

Step 2: Determination of mean by the direct method.

First, find the midpoint using the formula,

${\mathrm{x}}_{\mathrm{i}}=\frac{(\mathrm{upper}\mathrm{limit}+\mathrm{lower}\mathrm{limit})}{2}$

$\mathrm{Mean}=\mathrm{x̄}=\frac{\sum {\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}}{\sum {\mathrm{f}}_{\mathrm{i}}}$

$$\begin{gathered} =\frac{2830}{80} \\ =35.37\mathrm{years} \end{gathered}$$

$\Rightarrow \mathrm{Mean}=35.37\mathrm{years}$

Hence, the modal age is $36.8\mathrm{years}$which means the maximum number of patients admitted to the hospital are of age $36.8\mathrm{years}$ and the mean age is$35.37$ which means the average age of the patients admitted is $35.37\mathrm{years}.$