2. Quartile Example
1. Calculate Quartile-3 from the following grouped data
Solution:
| `x` | Frequency
`f` | `cf` | | 0 | 1 | 1 | | 1 | 5 | 6 | | 2 | 10 | 16 | | 3 | 6 | 22 | | 4 | 3 | 25 | | --- | --- | --- | | n = 25 | -- |
Here, `n = 25`
`Q_3 = ((3(n+1))/4)^(th)` value of the observation
`=((3*26)/4)^(th)` value of the observation
`=(19.5)^(th)` value of the observation
`=3`
2. Calculate Quartile-1 from the following grouped data
| X | Frequency | | 10 | 3 | | 11 | 12 | | 12 | 18 | | 13 | 12 | | 14 | 3 |
Solution:
| `x` | Frequency
`f` | `cf` | | 10 | 3 | 3 | | 11 | 12 | 15 | | 12 | 18 | 33 | | 13 | 12 | 45 | | 14 | 3 | 48 | | --- | --- | --- | | n = 48 | -- |
Here, `n = 48`
`Q_1 = ((n+1)/4)^(th)` value of the observation
`=(49/4)^(th)` value of the observation
`=(12.25)^(th)` value of the observation
`=11`
3. Calculate Quartile-3 from the following grouped data
| Class | Frequency | | 2 - 4 | 3 | | 4 - 6 | 4 | | 6 - 8 | 2 | | 8 - 10 | 1 |
Solution:
| Class | Frequency
`f` | `cf` | | 2 - 4 | 3 | 3 | | 4 - 6 | 4 | 7 | | 6 - 8 | 2 | 9 | | 8 - 10 | 1 | 10 | | --- | --- | --- | | n = 10 | -- |
Here, `n = 10`
`Q_3` class :
Class with `((3n)/4)^(th)` value of the observation in `cf` column
`=((3*10)/4)^(th)` value of the observation in `cf` column
`=(7.5)^(th)` value of the observation in `cf` column
and it lies in the class `6 - 8`.
`:. Q_3` class : `6 - 8`
The lower boundary point of `6 - 8` is `6`.
`:. L = 6`
`Q_3 = L + ((3 n)/4 - cf)/f * c`
`=6 + (7.5 - 7)/2 * 2`
`=6 + (0.5)/2 * 2`
`=6 + 0.5`
`=6.5`
4. Calculate Quartile-1 from the following grouped data
| Class | Frequency | | 0 - 2 | 5 | | 2 - 4 | 16 | | 4 - 6 | 13 | | 6 - 8 | 7 | | 8 - 10 | 5 | | 10 - 12 | 4 |
Solution:
| Class | Frequency
`f` | `cf` | | 0 - 2 | 5 | 5 | | 2 - 4 | 16 | 21 | | 4 - 6 | 13 | 34 | | 6 - 8 | 7 | 41 | | 8 - 10 | 5 | 46 | | 10 - 12 | 4 | 50 | | --- | --- | --- | | n = 50 | -- |
Here, `n = 50`
`Q_1` class :
Class with `(n/4)^(th)` value of the observation in `cf` column
`=(50/4)^(th)` value of the observation in `cf` column
`=(12.5)^(th)` value of the observation in `cf` column
and it lies in the class `2 - 4`.
`:. Q_1` class : `2 - 4`
The lower boundary point of `2 - 4` is `2`.
`:. L = 2`
`Q_1 = L + (( n)/4 - cf)/f * c`
`=2 + (12.5 - 5)/16 * 2`
`=2 + (7.5)/16 * 2`
`=2 + 0.9375`
`=2.9375`
5. Calculate Quartile-3 from the following grouped data
| Class | Frequency | | 10 - 20 | 15 | | 20 - 30 | 25 | | 30 - 40 | 20 | | 40 - 50 | 12 | | 50 - 60 | 8 | | 60 - 70 | 5 | | 70 - 80 | 3 |
Solution:
| Class | Frequency
`f` | `cf` | | 10 - 20 | 15 | 15 | | 20 - 30 | 25 | 40 | | 30 - 40 | 20 | 60 | | 40 - 50 | 12 | 72 | | 50 - 60 | 8 | 80 | | 60 - 70 | 5 | 85 | | 70 - 80 | 3 | 88 | | --- | --- | --- | | n = 88 | -- |
Here, `n = 88`
`Q_3` class :
Class with `((3n)/4)^(th)` value of the observation in `cf` column
`=((3*88)/4)^(th)` value of the observation in `cf` column
`=(66)^(th)` value of the observation in `cf` column
and it lies in the class `40 - 50`.
`:. Q_3` class : `40 - 50`
The lower boundary point of `40 - 50` is `40`.
`:. L = 40`
`Q_3 = L + ((3 n)/4 - cf)/f * c`
`=40 + (66 - 60)/12 * 10`
`=40 + (6)/12 * 10`
`=40 + 5`
`=45`
6. Calculate Quartile-1 from the following grouped data
| Class | Frequency | | 20 - 25 | 110 | | 25 - 30 | 170 | | 30 - 35 | 80 | | 35 - 40 | 45 | | 40 - 45 | 40 | | 45 - 50 | 35 |
Solution:
| Class | Frequency
`f` | `cf` | | 20 - 25 | 110 | 110 | | 25 - 30 | 170 | 280 | | 30 - 35 | 80 | 360 | | 35 - 40 | 45 | 405 | | 40 - 45 | 40 | 445 | | 45 - 50 | 35 | 480 | | --- | --- | --- | | n = 480 | -- |
Here, `n = 480`
`Q_1` class :
Class with `(n/4)^(th)` value of the observation in `cf` column
`=(480/4)^(th)` value of the observation in `cf` column
`=(120)^(th)` value of the observation in `cf` column
and it lies in the class `25 - 30`.
`:. Q_1` class : `25 - 30`
The lower boundary point of `25 - 30` is `25`.
`:. L = 25`
`Q_1 = L + (( n)/4 - cf)/f * c`
`=25 + (120 - 110)/170 * 5`
`=25 + (10)/170 * 5`
`=25 + 0.2941`
`=25.2941`
This material is intended as a summary. Use your textbook for detail explanation.
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