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Sagot :
Sure, let's break down the steps to create both a histogram and a frequency polygon based on the provided data.
### Given Data:
Class Intervals (C.I): [tex]\(0-6\)[/tex], [tex]\(6-12\)[/tex], [tex]\(12-18\)[/tex], [tex]\(18-30\)[/tex], [tex]\(30-36\)[/tex], [tex]\(36-42\)[/tex]
Frequencies (f): 4, 8, 15, 20, 12, 2
### Step 1: Construct the Histogram
1. Identify class intervals and frequencies:
- The class intervals are given as [tex]\(0-6\)[/tex], [tex]\(6-12\)[/tex], [tex]\(12-18\)[/tex], [tex]\(18-30\)[/tex], [tex]\(30-36\)[/tex] and [tex]\(36-42\)[/tex].
- The corresponding frequencies are 4, 8, 15, 20, 12, and 2.
2. Draw the axes:
- The horizontal axis (x-axis) will represent the class intervals.
- The vertical axis (y-axis) will represent the frequencies.
3. Plot the bars:
- Draw a bar for each class interval. The height of each bar corresponds to the frequency of the class interval.
- The width of each bar corresponds to the class interval width. In this case:
- [tex]\(0-6\)[/tex] has a frequency of 4
- [tex]\(6-12\)[/tex] has a frequency of 8
- [tex]\(12-18\)[/tex] has a frequency of 15
- [tex]\(18-30\)[/tex] has a frequency of 20
- [tex]\(30-36\)[/tex] has a frequency of 12
- [tex]\(36-42\)[/tex] has a frequency of 2
### Step 2: Construct the Frequency Polygon
1. Calculate Midpoints of Class Intervals:
- The midpoint [tex]\(M\)[/tex] for a class interval can be calculated as:
[tex]\[ M = \frac{\text{Lower class boundary} + \text{Upper class boundary}}{2} \][/tex]
- For each class interval:
- Midpoint of [tex]\(0-6\)[/tex]: [tex]\( \frac{0 + 6}{2} = 3 \)[/tex]
- Midpoint of [tex]\(6-12\)[/tex]: [tex]\( \frac{6 + 12}{2} = 9 \)[/tex]
- Midpoint of [tex]\(12-18\)[/tex]: [tex]\( \frac{12 + 18}{2} = 15 \)[/tex]
- Midpoint of [tex]\(18-30\)[/tex]: [tex]\( \frac{18 + 30}{2} = 24 \)[/tex]
- Midpoint of [tex]\(30-36\)[/tex]: [tex]\( \frac{30 + 36}{2} = 33 \)[/tex]
- Midpoint of [tex]\(36-42\)[/tex]: [tex]\( \frac{36 + 42}{2} = 39 \)[/tex]
2. Plot the points:
- Plot the points using the midpoints as [tex]\(x\)[/tex]-coordinates and frequencies as [tex]\(y\)[/tex]-coordinates.
- Points: (3, 4), (9, 8), (15, 15), (24, 20), (33, 12), (39, 2)
3. Draw the polygon:
- Connect the points with straight lines.
- For a complete polygon, you can optionally include the midpoints of the intervals before the first and after the last interval with frequency 0, but it's not mandatory.
### Summary
- Your histogram will consist of six bars with intervals on the x-axis and frequencies on the y-axis.
- Your frequency polygon will be a line graph connecting the points (3, 4), (9, 8), (15, 15), (24, 20), (33, 12), and (39, 2).
This method ensures you have a clear histogram and frequency polygon representing the distribution of frequencies across the class intervals.
### Given Data:
Class Intervals (C.I): [tex]\(0-6\)[/tex], [tex]\(6-12\)[/tex], [tex]\(12-18\)[/tex], [tex]\(18-30\)[/tex], [tex]\(30-36\)[/tex], [tex]\(36-42\)[/tex]
Frequencies (f): 4, 8, 15, 20, 12, 2
### Step 1: Construct the Histogram
1. Identify class intervals and frequencies:
- The class intervals are given as [tex]\(0-6\)[/tex], [tex]\(6-12\)[/tex], [tex]\(12-18\)[/tex], [tex]\(18-30\)[/tex], [tex]\(30-36\)[/tex] and [tex]\(36-42\)[/tex].
- The corresponding frequencies are 4, 8, 15, 20, 12, and 2.
2. Draw the axes:
- The horizontal axis (x-axis) will represent the class intervals.
- The vertical axis (y-axis) will represent the frequencies.
3. Plot the bars:
- Draw a bar for each class interval. The height of each bar corresponds to the frequency of the class interval.
- The width of each bar corresponds to the class interval width. In this case:
- [tex]\(0-6\)[/tex] has a frequency of 4
- [tex]\(6-12\)[/tex] has a frequency of 8
- [tex]\(12-18\)[/tex] has a frequency of 15
- [tex]\(18-30\)[/tex] has a frequency of 20
- [tex]\(30-36\)[/tex] has a frequency of 12
- [tex]\(36-42\)[/tex] has a frequency of 2
### Step 2: Construct the Frequency Polygon
1. Calculate Midpoints of Class Intervals:
- The midpoint [tex]\(M\)[/tex] for a class interval can be calculated as:
[tex]\[ M = \frac{\text{Lower class boundary} + \text{Upper class boundary}}{2} \][/tex]
- For each class interval:
- Midpoint of [tex]\(0-6\)[/tex]: [tex]\( \frac{0 + 6}{2} = 3 \)[/tex]
- Midpoint of [tex]\(6-12\)[/tex]: [tex]\( \frac{6 + 12}{2} = 9 \)[/tex]
- Midpoint of [tex]\(12-18\)[/tex]: [tex]\( \frac{12 + 18}{2} = 15 \)[/tex]
- Midpoint of [tex]\(18-30\)[/tex]: [tex]\( \frac{18 + 30}{2} = 24 \)[/tex]
- Midpoint of [tex]\(30-36\)[/tex]: [tex]\( \frac{30 + 36}{2} = 33 \)[/tex]
- Midpoint of [tex]\(36-42\)[/tex]: [tex]\( \frac{36 + 42}{2} = 39 \)[/tex]
2. Plot the points:
- Plot the points using the midpoints as [tex]\(x\)[/tex]-coordinates and frequencies as [tex]\(y\)[/tex]-coordinates.
- Points: (3, 4), (9, 8), (15, 15), (24, 20), (33, 12), (39, 2)
3. Draw the polygon:
- Connect the points with straight lines.
- For a complete polygon, you can optionally include the midpoints of the intervals before the first and after the last interval with frequency 0, but it's not mandatory.
### Summary
- Your histogram will consist of six bars with intervals on the x-axis and frequencies on the y-axis.
- Your frequency polygon will be a line graph connecting the points (3, 4), (9, 8), (15, 15), (24, 20), (33, 12), and (39, 2).
This method ensures you have a clear histogram and frequency polygon representing the distribution of frequencies across the class intervals.
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