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) × h`

           `⇒ 16 = 15 + ((70/2 −(24 + a))/ 15)× 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.