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Question
Calculate Q.D. of the following data:
| Height of plants (in feet) | 2 – 4 | 4 – 6 | 6 – 8 | 8 – 10 | 10 – 12 | 14 – 14 | 14 – 16 |
| No. of plants | 15 | 20 | 25 | 12 | 18 | 13 | 17 |
Solution
We construct the less than cumulative frequency table as follows:
| Height of plants (in feet) | No. of plants (f) |
Less than cumulative frequency (c.f.) |
| 2 – 4 | 15 | 15 |
| 4 – 6 | 20 | 35 ← Q1 |
| 6 – 8 | 25 | 60 |
| 8 – 10 | 12 | 72 |
| 10 – 12 | 18 | 90 ← Q3 |
| 12 – 14 | 13 | 103 |
| 14 – 16 | 17 | 120 |
| Total | N = 120 |
Here, N = 120
Q1 class = class containing `("N"/4)^"th"` observation
∴ `"N"/4 = 120/4` = 30
Cumulative frequency which is just greater than (or equal to) 30 is 35.
∴ Q1 lies in the class 4 – 6
∴ L = 4, c.f. = 15, f = 20, h = 2
∴ Q1 = `"L" + "h"/"f"("N"/4 - "c.f.")`
= `4 + 2/20 (30 - 15)`
= `4 + 1/10 xx 15`
= 4 + 1.5
= 5.5
Q3 class = class containing `((3"N")/4)^"th"` observation
∴ `(3"N")/4 = (3 xx 120)/4` = 90
Cumulative frequency which is just greater than (or equal to) 90 is 90.
∴ Q3 lies in the class 10 – 12
∴ L = 10, c.f. = 72, f = 18, h = 2
∴ Q3 = `"L" + "h"/"f"((3"N")/4 - "c.f.")`
= `10 + 2/18 (90 - 72)`
= `10 + 2/18 xx 18`
= 10 + 2
∴ Q3 = 12
∴ Q.D. = `("Q"_3 - "Q"_1)/2`
= `(12 - 5.5)/2`
= `(6.5)/2`
= 3.25
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