Question

If the median of the distribution given below is 28.5,

Class	0-10	10-20	20-30	30-40	40-50	50-60	Total
Freq.	5	a	20	15	b	5	60

then the missing value of a and b are respectively:
1. 3, 7
2. 2, 5
3. 8, 7
4. 8, 5

Answer 1

Correct Answer - Option 3 : 8, 7

Given:.

Median = 28.5

Number (N) = ∑ f_i = 60

Concept:

Cumulative frequency distribution is the sum of the frequency of class and all classes below it in a frequency distribution.

This means we can get cumulative frequency by adding up a value and all of the values that came before it.

Also,

Frequency of any class = cumulative frequency of class - cumulative frequency of preceding class.

\(Median= l + \frac{\frac{N}{2}-cf}{f}\)

\(l\) = lower limit of median class interval
cf = cumulative frequency preceding to the median class frequency
f = frequency of the class interval to which median belongs
h = width of the class interval

Solution:

Class interval	Frequency (fi)	Cumulative interval
0 - 10	5	5
10 - 20	a	5 + a
20 - 30	20	25 + a
30 - 40	15	40 + a
40 - 50	b	4 + a + b
50 - 60	5	45 + a + b
Total	∑fi = 60

 

 

 

 

 

 

 

 

From the Table,

Median Class = 20 - 30

N = ∑ f_i = 60

\(l\) = 20.

cf = 5 + x

f = 20

h = 10 - 0 = 10

Substituting in the formula, \(Median= l + \frac{\frac{N}{2}-cf}{f}\),

\(\Rightarrow28.5= 20+\frac{30 - (5-a)}{20}× 10\)

\(\Rightarrow28.5= 20+\frac{25-a}{2}\)

\(\Rightarrow28.5-20= \frac{25-a}{2}\)

\(\Rightarrow8.5= \frac{25-a}{2}\)

⇒ 8.5 × 2 = 25 - a

⇒ a = 25 - 17

⇒ a = 8

Substituting the value of a,

∑ f_i = 60 = 45 + a + b

⇒ 60 = 45 + 8 + b

⇒ b = 60 - 53

⇒ b = 7

Hence, a = 8, b = 7