Draw the histogram to represent the distribution.
Draw a frequency polygon.
Answer (1)
Calculate the midpoints of each interval: 24.5, 34.5, 44.5, 54.5, 62.
Adjust the height of the last rectangle in the histogram to 3.6.
Plot the points (24.5, 13), (34.5, 10), (44.5, 10), (54.5, 9), and (62, 18) for the frequency polygon.
Add points (14.5, 0) and (69.5, 0) to close the polygon.
Explanation
Analyze the problem and data We are given a frequency distribution table of marks obtained by 60 students. The table is as follows:
Marks
20-29
30-39
40-49
50-59
60-64
No. of students
13
10
10
9
18
We need to draw a histogram and a frequency polygon to represent this data.
Explain how to draw the histogram For the histogram, the x-axis represents the marks intervals, and the y-axis represents the number of students. The height of each bar corresponds to the number of students in that interval. The width of each bar corresponds to the interval size. Note that the interval 60-64 has a different width than the others, so the height of the bar needs to be adjusted to maintain the area proportional to the frequency.
Adjust the height of the last rectangle The class widths are not uniform. The first four classes have a width of 10, while the last class has a width of 5. To draw the histogram correctly, we need to adjust the heights of the rectangles so that the area of each rectangle is proportional to the frequency. For the class 60-64, the frequency density is calculated as follows:
Frequency density = Frequency / Class width Frequency density = 18 / 5 = 3.6
So, the height of the rectangle for the class 60-64 will be 3.6.
Calculate the midpoints of the intervals For the frequency polygon, we plot the midpoints of each interval against the number of students in that interval. The midpoints are calculated as follows:
20-29: (20 + 29) / 2 = 24.5 30-39: (30 + 39) / 2 = 34.5 40-49: (40 + 49) / 2 = 44.5 50-59: (50 + 59) / 2 = 54.5 60-64: (60 + 64) / 2 = 62
So, the points to be plotted are (24.5, 13), (34.5, 10), (44.5, 10), (54.5, 9), and (62, 18).
Add points to close the polygon To close the polygon, we add points at the midpoints of the intervals before the first and after the last, with a frequency of zero. These points are:
Before the first interval: (20 - 10) + 4.5 = 14.5. So the point is (14.5, 0). After the last interval: 60 + 4 + 5 = 69. So the point is (69.5, 0).
Connect the points with straight lines to complete the frequency polygon.
Examples
Histograms and frequency polygons are used in many real-world scenarios to visualize data distributions. For example, a teacher can use a histogram to visualize the distribution of test scores in a class. This can help the teacher identify areas where students are struggling and adjust their teaching accordingly. Similarly, a business can use a frequency polygon to visualize the distribution of sales data. This can help the business identify trends and make informed decisions about inventory and marketing.
