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Question
Weekly wages for a group of 100 persons are given below:
| Wages (in ₹) | 0 – 500 | 500 – 1000 | 1000 – 1500 | 1500 – 2000 | 2000 – 2500 |
| No. of persons | 7 | ? | 25 | 30 | ? |
D3 for this group is ₹ 1100 Calculate the missing frequencies.
Solution
Let a and b be the missing frequencies of the class 500 – 1000 and class 2000 – 2500 respectively.
We construct the less than cumulative frequency table as given below:
| Wages (in ₹) | No. of persons (f) | Less than Cumulative frequency (c.f.) |
| 0 – 500 | 7 | 7 |
| 500 – 1000 | a | 7 + a |
| 1000 – 1500 | 25 | 32 + a ← D3 |
| 1500 – 2000 | 30 | 62 + a |
| 2000 – 2500 | 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, D3 = 1100
∴ D3 lies in the class 1000 – 1500.
∴ L = 1000, h = 500, f = `(3"N")/10=(3xx100)/10` = 30,
c.f. = 7 + a
∴ D3 = `"L"+"h"/"f"((3"N")/(10) - "c.f.")`
∴ 1100 = `1000+(500)/(25)[30 - (7+"a")]`
∴ 1100 − 1000 = 20(30 – 7 – a)
∴ 100 = 20 (23 – a)
∴ 100 = 460 – 20a
∴ 20a = 460 – 100
∴ a = `360/20`
∴ 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 500 – 1000 and class 2000 – 2500 respectively.
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