Question

The median of the given data is 28.5.

Class	Frequency
0 – 10	5
10 – 20	a
20 – 30	20
30 – 40	15
40 – 50	b
50 – 60	5
Total	60

Find the values of a and b.

Answer 1

1. Construct the cumulative frequency table

The first step is to calculate the cumulative frequency (cf) for each class interval.

Class Interval	Frequency (f)	Cumulative Frequency (cf)
0 – 10	5	5
10 – 20	a	5 + a
20 – 30	20	25 + a
30 – 40	15	40 + a
40 – 50	b	40 + a + b
50 – 60	5	45 + a + b
Total	60

2. Formulate the total frequency equation

The total frequency is given as 60. From the table, the sum of all frequencies is 45 + a + b.

45 + a + b = 60

a + b = 15   ...(Equation 1)

3. Identify median class and Apply formula

The given median is 28.5, which falls within the class interval 20 – 30.

Thus, the median class is 20 – 30.

Lower limit (l): 20

Total frequency (n): 60 ⇒ `n/2 = 30`

Cumulative frequency of preceding class (cf): 5 + a

Frequency of median class (f): 20

Class width (h): 10

The formula for the median is:

Median = `l + ((n/2 - cf)/f) xx h`

Substitute the values:

`28.5 = 20 + ((30 - (5 + a))/20) xx 10`

`28.5 - 20 = (25 - a)/2`

8.5 × 2 = 25 – a

17 = 25 – a

a = 8

4. Solve for b

Using Equation 1:

8 + b = 15

b = 7

The values of the unknown frequencies are a = 8 and b = 7.