Question

Question 1
The following table shows the ages of the patients admitted in a hospital during a year:

$\begin{array}{ccccccc}\text{Age(in years)}& 5-15& 15-25& 25-35& 35-45& 45-55& 55-65\\ \text{Number of patients}& 6& 11& 21& 23& 14& 5\end{array}$

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

Answer 1

We may compute the class marks $\left(x_i\right)$ as per the relation:

$x_i=\frac{\text{Upper class limit+lower class limit}}{2}$

Now, taking 30 as assumed mean $a$, we may calculate $d_{i}and{f}_i{d}_i$ as following:

$\begin{array}{ccccc}\text{Age(in years)\&Number of patients}{f}_i& \text{class mark}{x}_i& d_i=x_i-30& f_i{d}_i& \\ 5-15& 6& 10& -20& -120\\ 15-25& 11& 20& -10& -110\\ 25-35& 21& 30& 0& 0\\ 35-45& 23& 40& 10& 230\\ 45-55& 14& 50& 20& 280\\ 55-65& 5& 60& 30& 150\\ \text{Total}& 80& & & 430\end{array}$

From the table, we may observe that:

$\sum f_i=80\sum f_i{d}_i=430Mean\overline{x}=a+\frac{\sum f_i{d}_i}{\sum f_i}=30+\left(\frac{430}{80}\right)$

= 30 + 5.375

= 35.375

Approximately equals to = 35.38

Clearly, mean of this data is 35.38. It represents that on an average the age of a patient admitted to hospital was 35.38 years.

As we may observe that maximum class frequency is 23 belonging to class interval 35 - 45.

So, modal class = 35 - 45

Lower limit $l$ of modal class = 35

Frequency $\left(f_1\right)$ of modal class = 23

Class size $h$ = 10

Frequency $\left(f_0\right)$ of class preceding the modal class = 21

Frequency $\left(f_2\right)$ of class succeeding the modal class = 14

$Nowmode=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h=35+\left(\frac{23-21}{2\left(23\right)-21-14}\right)\times 10=35+\left[\frac{2}{46-35}\right]\times 10=34+\frac{20}{11}$

= 35 + 1.81

= 36.8

Clearly, mode is 36.8. It represents that maximum number of patients admitted in hospital were of 36.8 years.