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 years): 5-15 15-25 25-35 35-45 45-55 55-65
No.of students: 6 11 21 23 14 5

Answer 1

Mode and media of the given data are 35.375 and 36.8 respectively.

Age                 No.of                         xi                         fi * xi

(in years)       students (fi)

5-15                       6                (5+15)/2 = 10             6 * 10 = 60

15-25                     11              (15+25)/2 = 20             11 * 20 = 220

25-35                    21              (25+35)/2 = 30            21 * 30 = 630

35-45                     23             (35+45)/2 = 40            23 * 40 = 920

45-55                     14              (45+55)/2 = 50             14 * 50 = 700

55-65                     5               (55+65)/2 = 60             5 * 60 = 300

                      ∑fi = 80                                                 ∑fi*xi = 2830

Mean = ∑fi*xi / ∑fi

= 2830 / 80

= 35.375

Modal class = class interval with highest frequency

35-45 with a highest frequency of 23

l = lower limit of modal class = 35

h = class interval = 15 - 5 = 10

f1 = frequency of modal class = 23

f0 = frequency of class before modal class = 21

f2 = frequency of class after modal class = 14

$Mode = l + \dfrac{f_1-f_0}{2f_1-f_0-f_2} * h$

Mode = 35 + [ (23-21) / (2*23 - 21 - 14) ] * 10

= 35 + 2/11 * 10

= 35 + 1.8

= 36.8

Answer 2

Answer:

Since the maximum frequency is 23 and it corresponds to the class 35−45.

∴ modal class =35−45

Here, x

k

=35,h=10,f

k

=23,f

k−1

=21,f

k+1

=14

We know that mode M

0

is given by

M

0

=x

k

+h

2f

k

−f

k−1

−f

k+1

f

k

−f

k−1

=35+10(

2(23)−21−14

23−21

)=35+

11

10(2)

=35+

11

20

=35+1.8=36.8 years nearly

Again, we have

Age (in years)

Number of patients (f

i

)

Mid-value (x

i

)

f

i

x

i

5−15

6

10

60

15−25

11

20

220

25−35

21

30

630

35−45

23

40

920

45−55

14

50

700

55−65

5

60

300

∑f

i

=80

∑f

i

x

i

=2830

Mean

x

=

∑f

i

∑f

i

x

i

=

80

2830

=

8

283

=35.37years

This shows that maximum number of patients admitted in the hospital are of the age 36.8 years while on an average the age of a patient admitted to the hospital is 35.37 years.