Draw a histogram and a frequency polygon of the frequency distribution
class 1-5,6-10,11-15
frequency 5,7,10
find the modal class,the mode,the mean
To find the modal class, mode, and mean of the given frequency distribution, we will first construct a histogram and a frequency polygon.
1. Histogram:
A histogram is a graphical representation of data that uses bars to represent the frequency distribution of a dataset.
First, determine the class intervals for the histogram based on the given frequency distribution:
- Class 1-5
- Class 6-10
- Class 11-15
The class intervals represent ranges of values, and the frequencies represent the number of observations within each class.
To draw the histogram, follow these steps:
- Draw a horizontal axis (x-axis) and label it with the class intervals (1-5, 6-10, 11-15).
- Draw a vertical axis (y-axis) and label it with the frequency values (0, 5, 10, 15, ...).
- Draw rectangles (bars) above each class interval on the x-axis, with the heights representing the frequencies.
- Ensure that the bars are of equal width and have no gaps between them.
Based on the given frequency distribution, the heights of the bars would correspond to the frequencies: 5, 7, and 10, respectively.
2. Frequency Polygon:
A frequency polygon is another graphical representation of data that uses a line to connect the midpoints of the bars in a histogram. It helps visualize the shape of the distribution.
To draw the frequency polygon, follow these steps:
- Use the midpoints of each class interval (3, 8, and 13) on the x-axis.
- Plot the midpoints on the x-axis and the corresponding frequencies on the y-axis.
- Connect the points with line segments.
- Extend the line segments to the vertical axis at the lower and upper limits of the frequency distribution.
Now that we have constructed the histogram and the frequency polygon, we can find the modal class, mode, and mean.
- Modal Class: The class with the highest frequency is the modal class. In this case, the class "11-15" has the highest frequency of 10.
- Mode: The mode is the value that appears most frequently in the dataset. Since the modal class is "11-15," we can consider the midpoint of this class (13) as the mode.
- Mean: To find the mean, we need to calculate the weighted average of the dataset.
The midpoints of the classes (3, 8, and 13) can be used as representative values for their respective intervals.
The formula to calculate the mean is:
Mean = (Sum of (midpoint * frequency)) / (Total frequency)
Mean = ((3 * 5) + (8 * 7) + (13 * 10)) / (5 + 7 + 10)
Calculating the values:
Mean = (15 + 56 + 130) / 22
Mean = 201 / 22
Mean ≈ 9.14
Therefore, the modal class is "11-15," the mode is 13, and the mean is approximately 9.14.