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Sagot :
Certainly! Let's go through each graphical representation step by step for the given data. The data tells us the number of students falling into different mark intervals.
### Data:
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|----------|------|-------|-------|-------|-------|-------|-------|
| Students | 5 | 10 | 20 | 30 | 12 | 8 | 5 |
### 1. Histogram
A histogram is a bar graph that represents the frequency of occurrence of different ranges of data. Here, it shows the number of students within each mark interval.
Steps to draw a histogram:
1. On the x-axis, plot the mark intervals.
2. On the y-axis, plot the number of students.
3. Draw a bar for each mark interval where the height of the bar corresponds to the number of students in that interval.
```
Marks (X-axis) 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
Number of Students (Y-axis)
| | | | | | |
| | | | | | |
| | | | | | |
30 | | ||| | |
| | ||| | |
20 | |||| | |
| |||| | |
10 ||||| | |
||||| ||
5 ||| | ||*|
```
### 2. Frequency Polygon
A frequency polygon is a graph that uses line segments to connect the midpoints of each interval in the histogram.
Steps to draw a frequency polygon:
1. Calculate the midpoints of each mark interval.
Midpoints: [5, 15, 25, 35, 45, 55, 65].
2. Plot the midpoints on the x-axis and the corresponding frequencies (number of students) on the y-axis.
3. Connect the points with line segments.
### 3. Cumulative Frequency Curve (Less than Ogive)
A less-than ogive (cumulative frequency curve) is a graph that shows the cumulative frequency for each class interval.
Steps to draw a less-than ogive:
1. Calculate the cumulative frequency:
- 5 (Score < 10)
- 15 (Score < 20), because 5 + 10 = 15
- 35 (Score < 30), because 15 + 20 = 35
- 65 (Score < 40), because 35 + 30 = 65
- 77 (Score < 50), because 65 + 12 = 77
- 85 (Score < 60), because 77 + 8 = 85
- 90 (Score < 70), because 85 + 5 = 90
2. Plot these cumulative frequencies at the upper class boundaries (10, 20, 30, 40, 50, 60, 70) on the x-axis against the cumulative frequency values on the y-axis.
### 4. Cumulative Frequency Curve (More than Ogive)
A more-than ogive is a graph that shows the cumulative frequency from the highest interval downward.
Steps to draw a more-than ogive:
1. Calculate the cumulative frequency from the highest interval downward.
- 90 (Score ≥ 0)
- 85 (Score ≥ 10), because 90 - 5 = 85
- 75 (Score ≥ 20), because 85 - 10 = 75
- 55 (Score ≥ 30), because 75 - 20 = 55
- 25 (Score ≥ 40), because 55 - 30 = 25
- 13 (Score ≥ 50), because 25 - 12 = 13
- 5 (Score ≥ 60), because 13 - 8 = 5
2. Plot these values at the lower class boundaries (0, 10, 20, 30, 40, 50, 60) on the x-axis against the cumulative frequency values on the y-axis.
### Combined Ogives
To visualize both less-than and more-than ogives on the same graph:
1. Use the midpoints on the x-axis.
2. Plot both the cumulative frequencies (less-than and more-than) accordingly.
By following these steps, you can create the requested graphs (Histogram, Frequency Polygon, and Cumulative Frequency Curves) using the given data.
### Data:
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|----------|------|-------|-------|-------|-------|-------|-------|
| Students | 5 | 10 | 20 | 30 | 12 | 8 | 5 |
### 1. Histogram
A histogram is a bar graph that represents the frequency of occurrence of different ranges of data. Here, it shows the number of students within each mark interval.
Steps to draw a histogram:
1. On the x-axis, plot the mark intervals.
2. On the y-axis, plot the number of students.
3. Draw a bar for each mark interval where the height of the bar corresponds to the number of students in that interval.
```
Marks (X-axis) 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
Number of Students (Y-axis)
| | | | | | |
| | | | | | |
| | | | | | |
30 | | ||| | |
| | ||| | |
20 | |||| | |
| |||| | |
10 ||||| | |
||||| ||
5 ||| | ||*|
```
### 2. Frequency Polygon
A frequency polygon is a graph that uses line segments to connect the midpoints of each interval in the histogram.
Steps to draw a frequency polygon:
1. Calculate the midpoints of each mark interval.
Midpoints: [5, 15, 25, 35, 45, 55, 65].
2. Plot the midpoints on the x-axis and the corresponding frequencies (number of students) on the y-axis.
3. Connect the points with line segments.
### 3. Cumulative Frequency Curve (Less than Ogive)
A less-than ogive (cumulative frequency curve) is a graph that shows the cumulative frequency for each class interval.
Steps to draw a less-than ogive:
1. Calculate the cumulative frequency:
- 5 (Score < 10)
- 15 (Score < 20), because 5 + 10 = 15
- 35 (Score < 30), because 15 + 20 = 35
- 65 (Score < 40), because 35 + 30 = 65
- 77 (Score < 50), because 65 + 12 = 77
- 85 (Score < 60), because 77 + 8 = 85
- 90 (Score < 70), because 85 + 5 = 90
2. Plot these cumulative frequencies at the upper class boundaries (10, 20, 30, 40, 50, 60, 70) on the x-axis against the cumulative frequency values on the y-axis.
### 4. Cumulative Frequency Curve (More than Ogive)
A more-than ogive is a graph that shows the cumulative frequency from the highest interval downward.
Steps to draw a more-than ogive:
1. Calculate the cumulative frequency from the highest interval downward.
- 90 (Score ≥ 0)
- 85 (Score ≥ 10), because 90 - 5 = 85
- 75 (Score ≥ 20), because 85 - 10 = 75
- 55 (Score ≥ 30), because 75 - 20 = 55
- 25 (Score ≥ 40), because 55 - 30 = 25
- 13 (Score ≥ 50), because 25 - 12 = 13
- 5 (Score ≥ 60), because 13 - 8 = 5
2. Plot these values at the lower class boundaries (0, 10, 20, 30, 40, 50, 60) on the x-axis against the cumulative frequency values on the y-axis.
### Combined Ogives
To visualize both less-than and more-than ogives on the same graph:
1. Use the midpoints on the x-axis.
2. Plot both the cumulative frequencies (less-than and more-than) accordingly.
By following these steps, you can create the requested graphs (Histogram, Frequency Polygon, and Cumulative Frequency Curves) using the given data.
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