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Question
Calculate Q.D. from the following data:
| X (less than) | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
| Frequency | 5 | 8 | 15 | 20 | 30 | 33 | 35 |
Solution
We construct the less than cumulative frequency table as follows:
| Class | f | Less than cumulative frequency (c.f.) |
| 0 – 10 | 5 | 5 |
| 10 – 20 | 3 | 8 |
| 20 – 30 | 7 | 15 ← Q1 |
| 30 – 40 | 5 | 20 |
| 40 – 50 | 10 | 30 ← Q3 |
| 50 – 60 | 3 | 33 |
| 60 – 70 | 2 | 35 |
| Total | N = 35 |
Here, N = 35
Q1 class = class containing `("N"/4)^"th"` observation
∴ `"N"/4 = 35/4` = 8.75
Cumulative frequency which is just greater than (or equal to) 8.75 is 15.
∴ Q1 lies in the class 20 – 30.
∴ L = 20, c.f. = 8, f = 7, h = 10
∴ Q1 = `"L" + "h"/"f"("N"/4 - "c.f.")`
= `20 + 10/7(8.75 - 8)`
= `20 + 10/7 xx 0.75`
= 20 + 1.07
∴ Q1 = 21.07
Q3 class = class containing `((3"N")/4)^"th"` observation
∴ `(3"N")/4 = (3 xx 35)/4` = 26.25
Cumulative frequency which is just greater than (or equal to) 26.25 is 30.
∴ Q3 lies in the class 40 – 50.
∴ L = 40, c.f. = 20, f = 10, h = 10
∴ Q3 = `"L" + "h"/"f"((3"N")/4 - "c.f.")`
= `40 + 10/10(26.25 - 20)`
= 40 + 6.25
∴ Q3 = 46.25
Q.D. = `("Q"_3 - "Q"_1)/2`
= `(46.25 - 21.07)/2`
= `(25.18)/2`
= 12.59
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