You can find the quartiles in a frequency distribution table by following these steps:
1. Calculate the Cumulative Frequency
Start by adding up the frequencies in each class interval. This gives you the cumulative frequency, which represents the total number of observations up to that particular class interval.
2. Determine the Position of Each Quartile
The first quartile (Q1) divides the data into the lowest 25% and highest 75%. The second quartile (Q2) is the median, dividing the data into the lowest 50% and highest 50%. The third quartile (Q3) divides the data into the lowest 75% and highest 25%.
You can find the position of each quartile using the following formulas:
- Q1 Position: (n + 1) / 4
- Q2 Position: (n + 1) / 2
- Q3 Position: 3(n + 1) / 4
Where 'n' is the total number of observations.
3. Locate the Quartile in the Cumulative Frequency Column
Identify the class interval where the calculated quartile position falls within the cumulative frequency.
4. Calculate the Quartile Value
Use the following formula to calculate the quartile value:
- *Quartile Value = Lower Limit of the Quartile Class + [(Quartile Position - Cumulative Frequency of the Preceding Class) / Frequency of the Quartile Class] Class Width**
Example:
Let's say you have a frequency distribution table for the heights of 50 students:
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 150-155 | 5 | 5 |
| 155-160 | 10 | 15 |
| 160-165 | 15 | 30 |
| 165-170 | 12 | 42 |
| 170-175 | 8 | 50 |
To find Q1:
- Q1 Position: (50 + 1) / 4 = 12.75
- Locate Q1: The Q1 position falls within the 155-160 class interval.
- Calculate Q1: Q1 = 155 + [(12.75 - 5) / 10] * 5 = 158.75 cm
Practical Insights:
- Quartiles are useful for understanding the spread and distribution of data.
- They can be used to identify outliers and to compare different datasets.
