Question

Find the mode for the following distribution.

Ages (in years)	1 – 10	11 – 20	21 – 30	31 – 40	41 – 50
No. of children	2	3	5	7	1

Answer 1

1. Make class boundaries continuous 

The original data is given in an discontinuous format (inclusive series). To analyze it accurately, we subtract 0.5 from each lower limit and add 0.5 to each upper limit:

Age Interval (Inclusive)	Continuous Class Boundaries	No. of Children (f)
1 – 10	0.5 – 10.5	2
11 – 20	10.5 – 20.5	3
21 – 30	20.5 – 30.5	5
31 – 40	30.5 – 40.5	7 (f1)
41 – 50	40.5 – 50.5	1

2. Identify the modal class variables

The highest frequency is 7, which belongs to the continuous class interval 30.5 – 40.5.

Lower boundary of the modal class (L) = 30.5

Frequency of the modal class (f₁) = 7

Frequency of the preceding class (f₀) = 5

Frequency of the succeeding class (f₂) = 1

Class width (h) = 40.5 – 30.5 = 10

3. Compute using the mode formula

Using the grouped data mode formula:

Mode = `L + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

Substitute the extracted values into the formula:

Mode = `30.5 + ((7 - 5)/(2(7) - 5 - 1)) xx 10`

Mode = `30.5 + (2/(14 - 6)) xx 10`

Mode = `30.5 + (2/8) xx 10`

Mode = 30.5 + 0.25 × 10

Mode = 30.5 + 2.5

Mode = 33.5