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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.
Solution
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.
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