OP Malhotra Class 9 Maths Solutions Chapter 15 Mean Median and Frequency Polygon Exercise 15 (C)

S Chand Class 9 ICSE Maths Solutions Chapter 15 Mean, Median and Frequency Polygon Ex 15(C)

Question 1. In a class of 60 boys, the marks obtained in a monthly test were as under :

Draw a frequency polygon to represents the above data.
Answer: First, we need to find the mid-point for each class interval. The mid-point is the average of the lower and upper limits of the class. Then we plot these mid-points against the number of students (frequency) on a graph. To properly close the frequency polygon, we add two extra class intervals, one before the first and one after the last, both with a frequency of zero. The midpoints are connected by straight lines.
Here is the table with mid-points:

Now, plot the points (15, 10), (25, 25), (35, 12), (45, 8) and (55, 5) on the graph. To complete the frequency polygon, add points (5, 0) and (65, 0) for the midpoints of the classes before and after the given range. Join all these points with straight lines to form the frequency polygon. This graph helps visualize how the marks are spread out in the class.

🎯 Exam Tip: Remember to always calculate the midpoint for each class interval (upper limit + lower limit)/2 before plotting. Also, close the frequency polygon by extending it to the midpoints of the adjacent zero-frequency class intervals on the x-axis.

Question 2. Represent the following data by frequency polygon?

Answer: To create a frequency polygon, we first determine the midpoint for each class interval in the given data. The midpoint is the central value of each mark range. Then, we plot these midpoints on the x-axis and their corresponding number of students (frequency) on the y-axis.
Here is the table with mid-points:

Next, we plot the points (5, 3), (15, 7), (25, 6), (35, 2), and (45, 5) on a graph sheet. To complete the frequency polygon, we connect these points with straight lines. We also connect the first point to (0,0) and the last point to (55,0) (midpoint of the next class 50-60 with zero frequency) on the x-axis, closing the polygon and making it touch the horizontal axis. This visual representation helps to see the distribution pattern of the data.

🎯 Exam Tip: Pay close attention to the starting and ending points of the polygon. For continuous data, the polygon typically touches the x-axis at the midpoints of the class intervals just before the first and after the last class with frequency zero.

Question 3.

Answer: To construct a frequency polygon from this data, we must first find the midpoint for each class interval. This midpoint represents the center of each class. Once we have the midpoints, we can plot them against their corresponding frequencies.
Here is the table with mid-points:

Now, plot the points (25, 7), (35, 3), (45, 5), (55, 2), and (65, 5) on the graph. To complete the frequency polygon, connect these points with straight lines. Also, connect the first point (25,7) to the midpoint of the previous class (15,0) and the last point (65,5) to the midpoint of the next class (75,0). This creates a closed shape that touches the x-axis, illustrating the distribution of the data.

🎯 Exam Tip: For class intervals that are not continuous (e.g., 20-29, 30-39), ensure you calculate the true class boundaries (e.g., 19.5-29.5) to find the correct midpoints, or apply a continuity correction if drawing a histogram before the polygon. The graph shown in the source uses the midpoints of the given intervals directly.

Question 4. Rohit asked people to draw a line 5 cm long using a straight edge without any markings on it. Here are the lengths in centimetres of the lines drawn : 4.3 3.2 3.9 4.7 5.8 6.1 5.7 6.2 6.5 3.7 4.2 5.1 6.5 7.2 7.4 3.7 5.8 4.2 4.1 5.0 5.1 4.7 3.2 3.5 5.2 2.9 2.8 4.3 5.1 4.8
(a) Draw up a grouped frequency table for the data. Use a class interval of 1 centimetre.
(b) Draw a frequency polygon for the data.
Answer: First, we need to organize the raw data into a grouped frequency table with a class interval of 1 cm. To do this, we find the smallest and largest values in the data set. The lowest length is 2.8 cm, and the highest length is 7.4 cm. These values help us set up the class intervals correctly. For each interval, we count how many lengths fall into it. We then find the midpoint for each interval, which is needed to draw the frequency polygon.
Here is the grouped frequency table:

For part (b), we plot the points from the table: (2.5, 2), (3.5, 6), (4.5, 8), (5.5, 8), (6.5, 4), and (7.5, 2). To draw the frequency polygon correctly, we need to add a point at the midpoint of the class before the first one (1.5, 0) and a point at the midpoint of the class after the last one (8.5, 0). Then, we connect all these points with straight lines to form the polygon, which shows the distribution of the line lengths.

🎯 Exam Tip: When creating a grouped frequency table, ensure the class intervals are consistent and cover the entire range of data. For a frequency polygon, remember to plot the midpoints of the class intervals and close the polygon on the x-axis.

Question 5. For the following data, draw a histogram and a frequency polygon.

Answer: To represent this data, we first draw a histogram and then superimpose a frequency polygon on it. A histogram uses bars to show frequency distribution, with the width of the bars representing the class interval and the height representing the frequency (number of persons).
Here is the table used for plotting:

To draw the histogram, we represent 'age' along the x-axis and 'number of persons' (frequency) along the y-axis. The bars for each age group are drawn touching each other because the data is continuous. For the frequency polygon, we find the mid-points of the top of each histogram bar. We then plot these mid-points and connect them with straight lines. To complete the polygon, we connect the first midpoint to the x-axis at the start of the first bar (0,0) and the last midpoint to the x-axis at the end of the last bar (42,0), as shown in the graph. This combined graph effectively shows both the rectangular representation of the histogram and the line graph of the frequency polygon.

🎯 Exam Tip: Ensure that the bars of the histogram are drawn touching each other for continuous data. When drawing the frequency polygon on top, make sure it starts and ends at the midpoints of the adjacent zero-frequency class intervals on the x-axis, or as depicted in the source graph.