Question

The median of the following data is 16. Find the missing frequencies a and b if the total of frequencies is 70.

Class	0 – 5	5 – 10	10 – 15	15 – 20	20 – 25	25 – 30	30 – 35	35 – 40
Frequency	12	a	12	15	b	6	6	4

Answer 1

Class	Frequency (f)	Cumulative Frequency (cf)
0 – 5	12	12
5 – 10	a	12 + a
10 – 15	12	24 + a
15 – 20	15	39 + a
20 – 25	b	39 + a + b
25 – 30	6	45 + a + b
30 – 35	6	51 + a + b
35 – 40	4	55 + a + b
Total	N = Σ𝑓𝑖 = 70

Let a and b be the missing frequencies of class intervals 5 – 10 and 20 – 25 respectively.

Then, 55 + a + b = 70 ⇒ a + b = 15    ...(1)

Median is 16, which lies in 15 – 20. So, the median class is 15 – 20.

∴ l = 15, h = 5, N = 70, f = 15 and cf = 24 + a

Now,

Median, `M = l + ((N/2 - cf)/f) xx h`

⇒ `16 = 15 + ((70/2 - (24 + a))/ 15) xx 5`

⇒ `16 = 15 + ((35 - 24 - a)/3)`

⇒ `16 = 15 + ((11 - a)/3)`

⇒ `16 - 15 = (11 - a)/3`

⇒ 1 × 3 = 11 – a

⇒ a = 11 – 3

⇒ a = 8

∴ b = 15 – a      [From (1)]

⇒ b = 15 – 8

⇒ b = 7

Hence, a = 8 and b = 7.