Question

Daily income for a group of 100 workers are given below:

Daily income (in₹)	0 – 50	50 – 100	100 – 150	150 – 200	200 – 250
No. of persons	7	?	25	30	?

P₃₀ for this group is ₹ 110. Calculate the missing frequencies.

Answer 1

Let a and b be the missing frequencies of the class 50 – 100 and class 200 – 250 respectively.
We construct the less than cumulative frequency table as given below:

Daily income (in ₹)	No. of persons (f)	Less than cumulative frequency (c.f.)
0 – 50	7	7
50 – 100	a	7 + a
100 – 150	25	32 + a ← P30
150 – 200	30	62 + a
200 – 250	b	62 + a + b
Total	62 + a + b

Here, N = 62 + a + b
Since, N = 100
∴ 62 + a+ b = 100
∴ a + b = 38  ..........(i)
Given, P₃₀ = 110
∴ P₃₀ lies in the class 100 – 150.

∴ L = 100, h = 50, f = 25, `(30"N")/100= (30 xx 100)/100` = 30, c.f. = 7 + a

∴ P₃₀ = `"L"+"h"/"f"((30"N")/100-"c.f.")`

∴ 110 = `100 + 50/25[30 - (7 + "a")]`

∴ 110 – 100 = 2(30 – 7 – a)
∴ 10 = 2(23 – a)

∴ `10/2` = 23 – a

∴ 5 = 23 – a
∴ a = 23 – 5
∴ a = 18
Substituting the value of a in equation (i), we get
18 + b = 38
∴ b = 38 – 18
∴ b = 20
∴ 18 and 20 are the missing frequencies of the class 50 – 100 and class 200 – 250 respectively.