Question

The median of the following data is 16. Find the missing frequencies a and b if the total of frequencies is 70.
$\begin{array}{ccccccccc}\text{Class}& 0-5& 5-10& 10-15& 15-20& 20-25& 25-30& 30-35& 35-40\\ \text{Frequency}& 12& a& 12& 15& b& 6& 6& 4\end{array}$

Answer 1

We prepare the cumulative frequency table, as shown:

Class	Frequency fi	Cumulative frequency
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,b be the missing frequencies of class intervals 5-10 and 20-25 respectively. Then,

55 + a + b = 70

⇒a+b=15 ........(i)

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

Therefore, 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

⇒1×3=11–a

⇒a=11–3

⇒a=8

Therefore,
b=15–a ( From (i) )

⇒b=15–8

⇒b=7

Hence, a= 8 and b = 7 .