Maharashtra State BoardHSC Commerce 11th
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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 - Mathematics and Statistics

Sum

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
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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

Concept: Measures of Dispersion - Quartile Deviation (Semi - Inter Quartile Range)
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