Question

The following table shows the ages of tha 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

To find the class mark ($x_i$) for each interval, the following relation is used.  

Class mark  $(x_i)$ = $\frac {\text{Upper \,limit + Lower \,limit}}{2}$

Taking 30 as assumed mean (a), $d_i$, and $f_id_i$ can be calculated as follows.   

Age (in years)	Number of patients $\bf{f_i}$	Class mark $\bf{x_i}$	$\bf{d_i = x_i -30}$	$\bf{f_id_i}$
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
Total	80			430

From the table, We obtain

$\sum f_i = 80$
$\sum f_id_i = 430$

Mean, $\overset{-}{x} = a + (\frac{\sum f_id_i}{\sum f_i})$

x = $30 + (\frac{430}{80})$

x = 30 + 5.375
x = 35.375
x = 35.38

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.

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It can be observed that the maximum class frequency is 23 belonging to class interval 35 - 45.
Modal class = 35 − 45 
Lower limit ($l$) of modal class = 35
Frequency ($f_1$) of modal class = 23
Class size ($h$) = 10  
Frequency ($f_0$) of class preceding the modal class = 21 
Frequency ($f_2$) of class succeeding the modal class = 14    

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

Mode = 3$5+ (\frac{23 - 21 }{ 2(23) - 21 - 14})$

Mode =$35 + [\frac{2}{46 - 35}] \times 10$

Mode = $35 + \frac{20}{ 11}$
Mode = 35 + 1.81
Mode = 36.8

Mode is 36.8. It represents that the age of maximum number of patients admitted in hospital was 36.8 years.