Question

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.

Answer 1

To find the class marks (x_i), the following relation is used.

`x_i  = ("Upper class limit + Lower class limit")/2`

Taking 30 as assumed mean (a), d_i and f_id_iare calculated as follows.

Age (in years)	Number of patients fi	Class mark xi	di = xi − 30	fidi
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:

`sumf_i = 80`

`sumf_i d_i = 430`

`"Mean" barx = a + ((sumf_iu_i)/(sumf_i))xh`

= `30 + (430/80)`

= 30 + 5.375

= 35.375

= 35.38

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

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 (f₁) of modal class = 23

Class size (h) = 10

Frequency (f₀) of class preceding the modal class = 21

Frequency (f₂) of class succeeding the modal class = 14

`Mode = l + ((f_1-f_0)/(2f_1-f_0-f_2)xh)`

= `35+((23-21)/(2(23)-21-14))xx10`

=`35+[2/(46-35)]xx10`

= `35+20/11`

= 36.8

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