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Question
The distribution of daily sales of shoes (size-wise) for 100 days from a certain shop is:
| Size of shoes | 2 | 4 | 3 | 5 | 7 | 6 | 8 |
| No. of days | 14 | 20 | 13 | 19 | 13 | 13 | 8 |
Compute Q2, D1, and P95.
Solution
By arranging the given data in ascending order, we construct the less than cumulative frequency table as given below:
| Size of shoes | No. of days (f) |
Less than cumulative frequency (c.f.) |
| 2 | 14 | 14 ← D1 |
| 3 | 13 | 27 |
| 4 | 20 | 47 |
| 5 | 19 | 66 ← Q2 |
| 6 | 13 | 79 |
| 7 | 13 | 92 |
| 8 | 8 | 100 ← P95 |
| Total | 100 |
Here, n = 100
Q2 = value of `2(("n"+1)/4)^"th"` observation
= value of `2((100+1)/4)^"th"` observation
= value of (2 × 25.25)th observation
= value of (50.5)th observation
Cumulative frequency, which is just greater than (or equal) to 50.5, is 66.
∴ Q2 = 5
D1 = value of `(("n"+1)/10)^"th"` observation
= value of `((100+1)/10)^"th"` observation
= value of (10.1)th observation
Cumulative frequency, which is just greater than (or equal) to 10.1, is 14.
∴ D1 = 2
P95 = value of `95(("n"+1)/10)^"th"` observation
= value of `95((100+1)/100)^"th"` observation
= value of (95 × 1.01)th observation
= value of (95.95)th observation
Cumulative frequency, which is just greater than (or equal) to 95.95 is 100.
∴ P95 = 8
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