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Calculate the 3rd quartile, 5th decile, and 90th percentile for the given grouped data.
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3 partsStep-by-step solution
9 stepsTo calculate the 3rd quartile, 5th decile, and 90th percentile for grouped data, we first need to determine the true class boundaries and cumulative frequencies.
The given data is:
| Score Interval | Number of Students () |
|---|---|
| 30-39 | 5 |
| 40-49 | 8 |
| 50-59 | 7 |
| 60-69 | 10 |
| 70-79 | 6 |
| 80-89 | 4 |
Step 1: Create a table with true class boundaries and cumulative frequencies.
The class width () is .
The total number of students () is the sum of frequencies: .
| Score Interval | True Class Boundaries | Frequency () | Cumulative Frequency () |
|---|---|---|---|
| 30-39 | 29.5 - 39.5 | 5 | 5 |
| 40-49 | 39.5 - 49.5 | 8 | 13 |
| 50-59 | 49.5 - 59.5 | 7 | 20 |
| 60-69 | 59.5 - 69.5 | 10 | 30 |
| 70-79 | 69.5 - 79.5 | 6 | 36 |
| 80-89 | 79.5 - 89.5 | 4 | 40 |
The formula for the -th percentile () is:
Where:
= lower class boundary of the percentile class
= total number of observations
= cumulative frequency of the class before the percentile class
= frequency of the percentile class
= class width
Part a· 3 steps
3rd quartile ()
Determine the position of .
The 3rd quartile is the 75th percentile ().
Position .Identify the 3rd quartile class.
The 30th observation falls in the class with a cumulative frequency of 30, which is 60-69 (true boundaries 59.5 - 69.5).
(cumulative frequency of the class before 60-69)
(frequency of the class 60-69)
Calculate .
The 3rd quartile is .
Part b
5th decile ()
Determine the position of .
The 5th decile is the 50th percentile ().
Position .Identify the 5th decile class.
The 20th observation falls in the class with a cumulative frequency of 20, which is 50-59 (true boundaries 49.5 - 59.5).
(cumulative frequency of the class before 50-59)
(frequency of the class 50-59)
Calculate .
The 5th decile is .
Part c
90th percentile ()
Determine the position of .
Position .
Identify the 90th percentile class.
The 36th observation falls in the class with a cumulative frequency of 36, which is 70-79 (true boundaries 69.5 - 79.5).
(cumulative frequency of the class before 70-79)
(frequency of the class 70-79)
Calculate .
The 90th percentile is .
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