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

We prepare the cumulative frequency table, as shown below:

Class	Frequency (fi)	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 = ∑fi = 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,

$$\begin{gathered} \mathrm{Median}=l+\left(\frac{{\displaystyle \frac{N}{2}}-cf}{f}\right)\times h \\ \Rightarrow 16=15+\left(\frac{{\displaystyle \frac{70}{2}}-\left(24+a\right)}{15}\right)\times 5 \\ \Rightarrow 16=15+\left(\frac{35-24-a}{3}\right) \\ \Rightarrow 16=15+\left(\frac{11-a}{3}\right) \\ \Rightarrow 16-15=\frac{11-a}{3} \\ \Rightarrow 1\times 3=11-a \\ \Rightarrow a=11-3 \\ \Rightarrow a=8 \end{gathered}$$

∴ b = 15 − a [From (1)]
⇒ b = 15 − 8
⇒ b = 7

Hence, a = 8 and b = 7.