Answer :
To draw a histogram and a frequency polygon for the given data set, follow these steps:
### Step 1: Organize the Data
You are given the following class intervals and their corresponding frequencies:
- Class Intervals: 20-25, 25-30, 30-35, 35-40, 40-45, 45-50
- Frequencies: 30, 24, 52, 28, 46, 10
### Step 2: Draw the Histogram
1. Draw the axes:
- The horizontal axis (x-axis) will represent the class intervals.
- The vertical axis (y-axis) will represent the frequency.
2. Draw the bars:
- For each class interval, draw a bar whose width corresponds to the class interval (e.g., 20-25) and height corresponds to the frequency (e.g., 30 for 20-25).
- Make sure there are no gaps between the bars since this is a continuous data set.
### Step 3: Calculate Midpoints for the Frequency Polygon
To draw a frequency polygon, calculate the midpoints of each class interval. The midpoint is the average of the lower and upper boundaries of the intervals:
- Midpoint for 20-25: [tex]\((20 + 25) / 2 = 22.5\)[/tex]
- Midpoint for 25-30: [tex]\((25 + 30) / 2 = 27.5\)[/tex]
- Midpoint for 30-35: [tex]\((30 + 35) / 2 = 32.5\)[/tex]
- Midpoint for 35-40: [tex]\((35 + 40) / 2 = 37.5\)[/tex]
- Midpoint for 40-45: [tex]\((40 + 45) / 2 = 42.5\)[/tex]
- Midpoint for 45-50: [tex]\((45 + 50) / 2 = 47.5\)[/tex]
### Step 4: Draw the Frequency Polygon
1. Plot the points:
- Use the calculated midpoints on the x-axis and the corresponding frequencies on the y-axis.
- For instance, plot a point at (22.5, 30) for the class interval 20-25.
2. Connect the points with lines:
- Start from the first midpoint (22.5, 30) to the last midpoint (47.5, 10), connecting all the points sequentially to form the frequency polygon.
3. Optional:
- You can extend the polygon to the x-axis by including a point at the beginning and the end of the interval range. For example, you can extend it by including points at (20, 0) and (50, 0) for completeness.
By following these steps, you can illustrate the distribution of the frequencies within each class interval using a histogram and visualize the overall shape of the data distribution with a frequency polygon.
### Step 1: Organize the Data
You are given the following class intervals and their corresponding frequencies:
- Class Intervals: 20-25, 25-30, 30-35, 35-40, 40-45, 45-50
- Frequencies: 30, 24, 52, 28, 46, 10
### Step 2: Draw the Histogram
1. Draw the axes:
- The horizontal axis (x-axis) will represent the class intervals.
- The vertical axis (y-axis) will represent the frequency.
2. Draw the bars:
- For each class interval, draw a bar whose width corresponds to the class interval (e.g., 20-25) and height corresponds to the frequency (e.g., 30 for 20-25).
- Make sure there are no gaps between the bars since this is a continuous data set.
### Step 3: Calculate Midpoints for the Frequency Polygon
To draw a frequency polygon, calculate the midpoints of each class interval. The midpoint is the average of the lower and upper boundaries of the intervals:
- Midpoint for 20-25: [tex]\((20 + 25) / 2 = 22.5\)[/tex]
- Midpoint for 25-30: [tex]\((25 + 30) / 2 = 27.5\)[/tex]
- Midpoint for 30-35: [tex]\((30 + 35) / 2 = 32.5\)[/tex]
- Midpoint for 35-40: [tex]\((35 + 40) / 2 = 37.5\)[/tex]
- Midpoint for 40-45: [tex]\((40 + 45) / 2 = 42.5\)[/tex]
- Midpoint for 45-50: [tex]\((45 + 50) / 2 = 47.5\)[/tex]
### Step 4: Draw the Frequency Polygon
1. Plot the points:
- Use the calculated midpoints on the x-axis and the corresponding frequencies on the y-axis.
- For instance, plot a point at (22.5, 30) for the class interval 20-25.
2. Connect the points with lines:
- Start from the first midpoint (22.5, 30) to the last midpoint (47.5, 10), connecting all the points sequentially to form the frequency polygon.
3. Optional:
- You can extend the polygon to the x-axis by including a point at the beginning and the end of the interval range. For example, you can extend it by including points at (20, 0) and (50, 0) for completeness.
By following these steps, you can illustrate the distribution of the frequencies within each class interval using a histogram and visualize the overall shape of the data distribution with a frequency polygon.
