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Question
Following data gives the weight of boxes.
Calculate Q.D. for the data:
| Weight (kg.) | 10 – 12 | 12 – 14 | 14 – 16 | 16 – 18 | 18 – 20 | 20 – 22 |
| No. of boxes | 3 | 7 | 16 | 14 | 18 | 2 |
| c.f. | 3 | 10 | 26 | 40 | 58 | 60 |
Solution
| Weight (kg.) | No. of boxes (f) |
c.f. |
| 10 – 12 | 3 | 3 |
| 12 – 14 | 7 | 10 |
| 14 – 16 | 16 | 26 ← Q1 |
| 16 – 18 | 14 | 40 |
| 18 – 20 | 18 | 58 ← Q3 |
| 20 – 22 | 2 | 60 |
| Total | N = 60 |
Here, N = 60
Q1 class = class containing `(("N")/4)^"th"` observation
∴ `"N"/(4) = (60)/(4)` = 15
Cumulative frequency which is just greater than (or equal to) 15 is 26.
∴ Q1 Lies in the class 14 – 16
∴ L = 14, f = 16, c.f. = 10, h = 2
∴ Q1 = `"L"+"h"/"f"("N"/4-"c.f.")`
= `14 + (2)/(16)(15 - 10)`
= `14 + 1/8xx5`
= 14 + 0.625
∴ Q1 = 14.625
Q3 class = class containing `((3"N")/4)^"th"` observation
∴ `(3"N")/(4) = (3xx60)/4` = 45
Cumulative frequency which is just greater than (or equal to) 45 is 58.
∴ Q3 Lies in the class 18 – 20
∴ L = 18, f = 18, c.f. = 40, h = 2
∴ Q3 = `"L"+"h"/"f"((3"N")/4-"c.f.")`
= `18 + (2)/(18)(45 - 40)`
= `18 + 1/9xx5`
= 18 + 0.5556
∴ Q3 = 18.5556
∴ Q.D. = `("Q"_3 - "Q"_1)/2`
= `(18.5556 - 14.625)/2`
= `3.9306/2`
= 1.9653
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