Identify the lower class limits, upper-class limits, class width, class midpoints, and class boundaries for the given frequency distribution. Also, identify the number of individuals included in the summary
The Correct Answer and Explanation is :
To analyze the given frequency distribution, we will determine the lower class limits, upper class limits, class width, class midpoints, class boundaries, and the total number of individuals included in the summary. The provided frequency distribution is as follows:
| Class Interval | Frequency |
|---|---|
| 25–34 | 27 |
| 35–44 | 35 |
| 45–54 | 15 |
| 55–64 | 3 |
| 65–74 | 5 |
| 75–84 | 1 |
| 85–94 | 2 |
1. Lower Class Limits:
The lower class limits are the smallest values in each class interval.
- 25, 35, 45, 55, 65, 75, 85
2. Upper Class Limits:
The upper class limits are the largest values in each class interval.
- 34, 44, 54, 64, 74, 84, 94
3. Class Width:
The class width is the difference between the lower class limit of one class and the lower class limit of the next class.
- Class width = 35 – 25 = 10
4. Class Midpoints:
The class midpoint is the average of the lower and upper class limits for each class interval.
- Midpoint of 25–34 = (25 + 34) / 2 = 29.5
- Midpoint of 35–44 = (35 + 44) / 2 = 39.5
- Midpoint of 45–54 = (45 + 54) / 2 = 49.5
- Midpoint of 55–64 = (55 + 64) / 2 = 59.5
- Midpoint of 65–74 = (65 + 74) / 2 = 69.5
- Midpoint of 75–84 = (75 + 84) / 2 = 79.5
- Midpoint of 85–94 = (85 + 94) / 2 = 89.5
5. Class Boundaries:
Class boundaries are the values that separate one class interval from another. They are calculated by subtracting 0.5 from the lower class limit and adding 0.5 to the upper class limit.
- Lower boundaries: 24.5, 34.5, 44.5, 54.5, 64.5, 74.5, 84.5
- Upper boundaries: 34.5, 44.5, 54.5, 64.5, 74.5, 84.5, 94.5
6. Number of Individuals Included in the Summary:
The total number of individuals is the sum of the frequencies.
- Total = 27 + 35 + 15 + 3 + 5 + 1 + 2 = 88
Summary Table:
| Class Interval | Frequency | Midpoint | Lower Boundary | Upper Boundary |
|---|---|---|---|---|
| 25–34 | 27 | 29.5 | 24.5 | 34.5 |
| 35–44 | 35 | 39.5 | 34.5 | 44.5 |
| 45–54 | 15 | 49.5 | 44.5 | 54.5 |
| 55–64 | 3 | 59.5 | 54.5 | 64.5 |
| 65–74 | 5 | 69.5 | 64.5 | 74.5 |
| 75–84 | 1 | 79.5 | 74.5 | 84.5 |
| 85–94 | 2 | 89.5 | 84.5 | 94.5 |
Explanation:
In a frequency distribution, class intervals are used to group data into ranges. The lower and upper class limits define the boundaries of these intervals. The class width is the difference between the lower class limits of two consecutive intervals, providing insight into the range of data within each class. Class midpoints represent the central value of each class interval and are calculated by averaging the lower and upper class limits. Class boundaries are the precise values that separate one class interval from another, ensuring that data points are accurately assigned to their respective classes without overlap. The total number of individuals is the sum of the frequencies across all class intervals, representing the entire dataset.
Understanding these components is crucial for analyzing and interpreting frequency distributions, as they provide a structured way to summarize and describe the distribution of data.