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Question
Calculate D9 and P20 of the following distribution.
| Length (in inches) | 0 – 20 | 20 – 40 | 40 – 60 | 60 – 80 | 80 – 100 | 100 – 120 |
| No. of units | 1 | 14 | 35 | 85 | 90 | 15 |
Solution
We construct the less than cumulative frequency table as given below:
| Length (in inches) | No. of units (f) | Less than Cumulative frequency (c.f.) |
| 0 – 20 | 1 | 1 |
| 20 – 40 | 14 | 15 |
| 40 – 60 | 35 | 50 ← P20 |
| 60 – 80 | 85 | 135 |
| 80 – 100 | 90 | 225 ← D9 |
| 100 – 120 | 15 | 240 |
| Total | 240 |
Here, N = 240
D9 class = class containing `((9"N")/10)^"th"` observation
∴ `(9"N")/(10) = (9 xx 240)/(10)` = 216
Cumulative frequency which is just greater than (or equal to) 216 is 225
∴ D9 lies in the class 80 – 100.
∴ L = 80, f = 90, c.f. = 135, h = 20
∴ D9 = `"L"+"h"/"f"((9"N")/(10) - "c.f.")`
= `80 + (20)/(90)(216 - 135)`
= `80+2/9(81)`
= 80 + 18
∴ D9 = 98
P20 class = class containing `((20"N")/100)^"th"` observation
∴ `(20"N")/(100) = (20 xx 240)/(100)` = 48
Cumulative frequency which is just greater than (or equal to) 48 is 50.
∴ P20 lies in the class 40 – 60
∴ L = 40, f = 35, c.f. = 15, h = 20
P20 = `"L"+"h"/"f"((20"N")/(100) - "c.f.")`
= `40 + (20)/(35)(48 - 15)`
= `40 + (4)/(7)(33)`
= `40+132/7`
∴ P20 = 58.86
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