OP Malhotra Class 10 Maths Solutions Chapter 19 Histogram and Ogive Exercise 19 (C)

Exercise 19(C)

 

Question 1. The daily wages of casual labour employed by a group of limited concerns are given below: Draw a cumulative frequency curve for the above data.
Answer: To draw a cumulative frequency curve (ogive), first, we need to find the cumulative frequency for the given data.

Next, we plot the points for the upper class boundaries and their corresponding cumulative frequencies. The points to plot are (5, 7), (7, 17), (9, 40), (11, 91), (13, 97), and (15, 100). We also consider the lower boundary of the first class interval with a cumulative frequency of 0, which is (3, 0). Then, we join these points with a free-hand curve to create the ogive.

In simple words: First, add up the frequencies to get cumulative frequencies. Then, on a graph, plot points using the upper limit of each wage group and its cumulative frequency. Start the curve from the lower limit of the very first group at zero frequency. Connect these points smoothly to draw the ogive.

🎯 Exam Tip: Always plot the upper class limits against the cumulative frequency for an 'less than' ogive. Remember to start the curve from the lower class limit of the first interval on the x-axis, with a cumulative frequency of zero.

 

Question 2. Draw a cumulative frequency curve for the following data :
(a)
(b)
(c)
Answer:
(a) To draw the cumulative frequency curve (ogive) for the marks data, we first calculate the cumulative frequencies:

We then plot the points corresponding to the upper class boundaries and their cumulative frequencies: (10, 5), (20, 15), (30, 37), (40, 77), (50, 92), and (60, 100). The curve starts from the lower limit of the first class interval, (0, 0). These points are joined by a free-hand curve to form the ogive.

(b) For the second set of data, we calculate the cumulative frequencies and then plot the ogive.We plot the points using the upper class boundaries and their cumulative frequencies: (5,20), (10,43), (15,68), (20,96), (25,126), (30,147), (35,175), and (40,191). The curve starts at (0,0), which is the lower limit of the first class interval with zero cumulative frequency. A smooth, free-hand curve connects these points to form the ogive.

(c) For data with inclusive class intervals, we first convert them to continuous class intervals and then calculate the cumulative frequencies.
We plot the points using the upper limits of the actual class intervals and their cumulative frequencies: (19.5, 10), (29.5, 25), (39.5, 45), (49.5, 70), (59.5, 100), and (69.5, 135). The curve begins from the lower limit of the first continuous class interval, (9.5, 0). These points are connected with a free-hand curve to create the ogive.

In simple words: To draw an ogive, first organize the data into groups and count how many times each group appears (frequency). Then, add these frequencies up for each group to get the cumulative frequency. Finally, plot points on a graph: for each group, use its upper boundary on the horizontal line and its cumulative frequency on the vertical line. Connect these points with a smooth line, starting from the lower boundary of the first group at zero.

🎯 Exam Tip: When class intervals are inclusive (like 10-19), remember to adjust them to be continuous (like 9.5-19.5) before calculating cumulative frequencies and plotting the ogive. This ensures the curve is accurate.

 

Question 3. Each of the 25 students in a class was given a home assignment comprising 10 questions in mathematics. The data given below show the number of questions solved and submitted by individual students on the next day. 1, 4, 5, 6, 0, 9, 3, 2, 3, 4, 6, 4, 5, 2, 7, 5, 2, 2, 3, 5, 1, 0, 7, 6, 3 (a) Taking classes as 0-2, 2-4, 4-6, etc., make a frequency table for the above distribution. (b) Draw an Ogive (cumulative frequency curve) to represent the given data.
Answer:
(a) To create the frequency table, we count how many students fall into each class interval (0-2, 2-4, etc.) and then calculate the cumulative frequency.

(b) To draw the ogive, we plot the upper class limits against their cumulative frequencies: (2, 7), (4, 12), (6, 19), (8, 24), and (10, 25). The curve starts from (0, 0), which is the lower limit of the first class interval with zero cumulative frequency. A smooth, free-hand curve is then drawn to connect these points, forming the ogive.

In simple words: First, sort the numbers into groups and count how many fall into each group. Then, add these counts to get cumulative frequencies. Finally, plot points on a graph where the horizontal line shows the group limits and the vertical line shows the cumulative counts. Connect these points with a smooth line to make the curve.

🎯 Exam Tip: When given raw data, carefully tally each value into its correct class interval. Make sure to use the upper class boundaries for plotting a 'less than' ogive, and start at zero for the lower boundary of the first class.

 

Question 4. Draw an ogive from the following table :
Answer: The given table shows "more than" cumulative frequencies. To draw a "more than" ogive, we plot the lower class limits against their corresponding cumulative frequencies.

The points to plot are (0, 100), (10, 100), (20, 96), (30, 68), (40, 26), and (50, 6). We start plotting from the lowest limit (0) with the total cumulative frequency, and then connect the points with a free-hand curve to form the "more than" ogive. This curve will typically slope downwards.

In simple words: When the table shows how many are "more than" a certain value, you're making a "more than" ogive. Plot the lowest value (like 0 marks) with the total count, then each value (like 10 marks) with its "more than" count. Connect these points with a smooth line, which will go downwards. This helps to see how many people scored above a certain mark.

🎯 Exam Tip: For a "more than" ogive, always plot the lower class limits against their respective cumulative frequencies. The curve should generally start at the lowest possible value with the maximum cumulative frequency and slope downwards.

 

Question 5. The marks secured by 50 students were as under: 14, 12, 18, 11, 25, 32, 27, 28, 27, 3, 9, 31, 5, 22, 13, 5, 22, 14, 10, 18, 40, 23, 28, 19, 18, 2, 13, 14, 22, 7, 46, 12, 36, 14, 17, 19, 35, 9, 12, 2, 49, 43, 7, 6, 10, 22, 3, 27 Taking the size of each class-interval as 10, prepare frequency table and with its help draw a cumulative frequency curve.
Answer: First, we identify the highest and lowest marks to determine the range of the data. The highest mark is 49 and the lowest is 2. Since the class interval size is 10, we can form class intervals like 0-10, 10-20, and so on.

To draw the ogive, we plot the upper class boundaries against their cumulative frequencies: (10, 11), (20, 30), (30, 41), (40, 46), and (50, 50). The curve starts from the lower limit of the first class interval, (0, 0). These points are then joined by a free-hand curve to form the ogive.

In simple words: First, find the lowest and highest marks. Then, group the marks into intervals of 10 and count how many students fall into each. Add these counts to get the running total (cumulative frequency). Finally, draw a graph by plotting the upper limit of each mark group against its running total, starting from zero for the first group's lower limit, and connect them with a smooth line. This helps you visualize how many students scored below a certain mark.

🎯 Exam Tip: Always specify the highest and lowest values in the raw data to ensure that your class intervals cover the entire range. Double-check your tally marks and cumulative frequency calculations to avoid errors in the final ogive.

 

Question 6. What is an Ogive curve? How is it useful? A group of 140 workers in a factory were given a work aptitude test. The distribution of their scores is given below :
Answer: An ogive curve is a special type of graph that shows the cumulative frequency distribution. It is drawn by plotting the cumulative frequencies against the upper or lower boundaries of the class intervals and connecting the points with a smooth curve. Ogives are useful for quickly estimating values like the median, quartiles, and percentiles from a dataset, or to find how many data points are above or below a certain value.

To draw the ogive, we plot the upper class limits against their cumulative frequencies: (15, 2), (20, 9), (25, 18), (30, 33), (35, 59), (40, 78), (45, 105), (50, 125), (55, 136), and (60, 140). The curve starts from the lower limit of the first class interval, (10, 0). These points are then joined by a free-hand curve to form the ogive.

In simple words: An ogive is a line graph that shows running totals of data. It helps you quickly find things like the middle value (median) or how many items are above or below a certain point. To draw it, first list scores and their frequencies, then add frequencies to get cumulative frequencies. Plot these cumulative numbers against the upper limit of each score group and connect them with a smooth line. This curve shows how the total count changes over the scores.

🎯 Exam Tip: When defining an ogive, always mention that it's a cumulative frequency curve. Highlight its primary uses, such as finding the median, quartiles, and the number of observations above or below a certain value, as these are key points an examiner looks for.

S Chand Class 10 ICSE Maths Solutions Chapter 19 Histogram And Ogive Ex 19(C)

 

Question 1. The daily wages of casual labour employed by a group of limited concerns are given below:

Draw a cumulative frequency curve for the above data.

Answer: First, we need to create a cumulative frequency table from the given data. To do this, we add up the frequencies one by one. For an ogive, we plot the upper class limits against the cumulative frequencies.Now, we plot the points using the upper class limit and the cumulative frequency: (5, 7), (7, 17), (9, 40), (11, 91), (13, 97), and (15, 100). We join these points with a smooth curve using freehand to create the ogive, also known as a cumulative frequency curve. This type of graph helps visualize how many data points fall below a certain value.
In simple words: First, list all the daily wages and count how many times each one appears. Then, make a new column where you keep adding up the counts as you go down the list. After that, draw a graph with two lines, one for daily wages and one for the total counts. Put dots on the graph at the end of each wage group and its total count, then connect these dots with a smooth, curved line. This curve shows how many workers earn less than a certain amount.

🎯 Exam Tip: Always remember to plot the upper class boundary against the cumulative frequency when drawing a "less than" ogive. Make sure your curve is smooth and does not have sharp corners.

 

Question 2. Draw a cumulative frequency curve for the following data :
(a)

(b)

(c)

Answer:
(a) To draw the cumulative frequency curve for this data, we first need to calculate the cumulative frequencies.Next, we plot the points using the upper class limit and the cumulative frequency: (10, 5), (20, 15), (30, 37), (40, 77), (50, 92) and (60, 100). We then connect these points with a smooth, freehand curve to form the ogive. This graph shows the total number of students who scored less than a particular mark.
In simple words: First, create a table to show how many students got marks up to each class limit. Then, draw a graph with marks on the bottom line and these total counts on the side line. Mark points for the upper limit of each mark group and its total count. Connect these dots with a smooth, curving line to get the graph.

🎯 Exam Tip: When class intervals are continuous (like 0-10, 10-20), use the upper limit for plotting. Ensure the axes are clearly labeled with appropriate scales for easy reading.

(b) For this part, we also need to calculate the cumulative frequencies first. The class intervals are already continuous.We plot the points using the upper class limit and cumulative frequency: (5,20), (10,43), (15,68), (20,96), (25,126), (30,147), (35,175) and (40,191). Then we join these points with a freehand curve to get the ogive. This curve graphically shows how the cumulative frequency changes with the class intervals, which can be useful for finding medians.
In simple words: First, create a list that shows the total number of items up to each class group. Then, draw a graph using the end value of each class group and its total count. Mark these points on the graph and connect them with a smooth, curving line by hand. This shows how the total count adds up over the groups.

🎯 Exam Tip: When dealing with discrete data or non-continuous intervals, ensure you adjust the class boundaries to be continuous before calculating cumulative frequency and plotting. Also, use a pencil for the freehand curve to ensure it is smooth.

(c) For part (c), the class intervals are not continuous (e.g., 10-19, 20-29). We need to make them continuous by finding the actual class intervals, which involves subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.Now, we plot the points using the upper actual class limit and cumulative frequency: (19.5, 10), (29.5, 25), (39.5, 45), (49.5, 70), (59.5, 100) and (69.5, 135). We then connect these points with a smooth, freehand curve to form the ogive. Adjusting class intervals helps make the graph accurate.
In simple words: First, adjust the class groups so they flow smoothly from one to the next (e.g., 9.5-19.5 instead of 10-19). Then, make a new list that shows the running total of frequencies for these new groups. After that, plot dots on a graph using the upper limit of each smooth group and its running total. Finally, draw a smooth, curved line connecting all these dots.

🎯 Exam Tip: Always check if class intervals are continuous. If they are not, like 10-19 and 20-29, convert them to continuous intervals (e.g., 9.5-19.5 and 19.5-29.5) before calculating cumulative frequencies and plotting the ogive. This ensures accuracy in the graph.

 

Question 3. Each of the 25 students in a class was given a home assignment comprising 10 questions in mathematics. The data given below show the number of questions solved and submitted by individual students on the next day.
1, 4, 5, 6, 0, 9, 3, 2, 3, 4, 6, 4, 5, 2, 7, 5, 2, 2, 3, 5, 1, 0, 7, 6, 3
(a) Taking classes as 0-2, 2-4, 4-6, etc., make a frequency table for the above distribution.
(b) Draw an Ogive (cumulative frequency curve) to represent the given data.

Answer:
(a) First, we organize the raw data into a frequency table with tally marks, using the given class intervals. We count how many students fall into each mark range and then calculate the cumulative frequency by adding up the counts.
(b) To draw the ogive, we plot the points using the upper class limit and cumulative frequency from the table: (2, 7), (4, 12), (6, 19), (8, 24), and (10, 25). These points are then connected with a smooth, freehand curve to create the ogive. This curve helps visualize the number of students who completed less than a certain number of questions.
In simple words: For part (a), count how many times each number of solved questions shows up in the given ranges and write them down. For part (b), draw a graph where the number of questions is on the bottom line and the total counts are on the side. Mark dots at the end of each question range and its total count, then draw a smooth, curved line to connect these dots. This graph helps see how many students answered less than a certain number of questions.

🎯 Exam Tip: When making a frequency table from raw data, always ensure to properly sort the data and accurately count tallies for each class interval. For drawing the ogive, remember to plot the upper limit of the class interval against its cumulative frequency.

 

Question 4. Draw an ogive from the following table :

Answer: This table shows a "more than" cumulative frequency distribution. To draw its ogive, we will plot the lower class limits against the cumulative frequencies given. The points to be plotted are (0, 100), (10, 96), (20, 68), (30, 26), and (40, 6). The ogive for "more than" data usually slopes downwards.
In simple words: Look at the table that shows how many items are "more than" a certain mark. Draw a graph using the lowest value of each mark group and its matching "more than" total. Mark these points and connect them with a smooth, curved line. This graph will show how many items are above a specific mark.

🎯 Exam Tip: For "more than" ogives, plot the lower class boundary against the cumulative frequency, and the curve will typically slope downwards. Always make sure to include a point at the highest class boundary with a cumulative frequency of zero for completion.

 

Question 5. The marks secured by 50 students were as under:
14, 12, 13, 18, 11, 25, 32, 27, 28, 27, 3, 9, 37, 31, 5, 22, 13, 5, 22, 14, 10, 18, 40, 23, 28, 19, 18, 2, 13, 14, 22, 7, 46, 12, 36, 14, 17, 19, 35, 9, 12, 2, 49, 43, 7, 6, 10, 22, 3, 27
Taking the size of each class-interval as 10, prepare frequency table and with its help draw a cumulative frequency curve.

Answer: First, we find the highest and lowest marks to determine the range for our class intervals. The highest mark is 49, and the lowest is 2. With a class interval size of 10, we can set up the class intervals. Then, we prepare a frequency table by tallying the marks in each interval and calculating the cumulative frequency.Now, we plot the points using the upper class limit and cumulative frequency: (10, 11), (20, 30), (30, 41), (40, 46) and (50, 50). We join these points with a freehand curve to create the ogive. This graph gives a visual summary of the students' marks, showing how many students scored below a specific mark.
In simple words: First, find the highest and lowest marks, then make groups of marks (class intervals) of size 10. Next, count how many students fall into each group and keep a running total. Finally, draw a graph by marking points for the top number of each mark group and its running total, then draw a smooth, curvy line through these points. This shows the total number of students scoring below a certain mark.

🎯 Exam Tip: Always calculate the range (highest minus lowest value) and then decide appropriate class intervals. Double-check your tally counts and cumulative frequencies, as any error will propagate through the ogive.

 

Question 6. What is an Ogive curve? How is it useful? A group of 140 workers in a factory were given a work aptitude test. The distribution of their scores is given below :

Draw an Ogive curve.

Answer: An ogive curve is a graph that shows the cumulative frequency distribution. It is drawn by plotting points where the x-axis represents the upper class limits of the data and the y-axis represents the cumulative frequencies. This graph is very useful for finding the median of the data, determining quartiles, and estimating the number of observations below or above a certain value.

To draw the ogive for the given data, we first need to calculate the cumulative frequencies:

Then, we plot the points (15, 2), (20, 9), (25, 18), (30, 33), (35, 59), (40, 78), (45, 105), (50, 125), (55, 136) and (60, 140) on the graph. These points represent the upper limit of each score interval and its corresponding cumulative frequency. We connect these points with a freehand smooth curve to create the ogive. This curve helps quickly see how many workers scored below a specific aptitude level.
In simple words: An ogive curve is a special graph that shows how many data points are below a certain value. It helps us easily find the middle value (median) or how many people fall into a certain group. To draw it, we first list the total count for each score group. Then, we plot these totals on a graph against the highest score in each group. We connect these dots with a smooth, curving line to see the full picture.

🎯 Exam Tip: Clearly define "Ogive" and explain its uses (like finding median, quartiles) at the beginning of the answer. When drawing, ensure the x-axis represents the upper class limits and the y-axis represents the cumulative frequency, and always connect points with a smooth, freehand curve.