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

Question 1. Represent the following distribution of ages (in years) of 35 teachers in a school by means of a histogram.

Answer: To make the histogram, we draw the x-axis for age (in years) and the y-axis for the number of teachers (which are the frequencies). Then, we create bars for each age group, with the height of each bar matching the number of teachers in that group. The bars should touch each other because the data is continuous. Here is how the histogram looks:

In simple words: We draw bars to show how many teachers are in each age group. The bottom line shows the age ranges, and the side line shows how many teachers are in each group. Each bar stands for an age group, and its height shows the count of teachers.

🎯 Exam Tip: Always ensure the bars in a histogram touch each other when representing continuous data, unlike a bar graph where they are separate. Label both axes clearly with units.

 

Question 2. The weekly observatios on cost of living index in a certain city for a year give the following frequency table :

Answer: To represent this data as a histogram, we will place the 'Cost of living index' values along the x-axis and the 'Number of weeks' (which are the frequencies) along the y-axis. The bars are drawn for each index range, with their heights showing how many weeks fall into that range. This visual representation helps to easily see the distribution of the cost of living index over the year. Here is the histogram:

In simple words: This picture shows how often different costs of living happened each week. The bottom line tells you the cost ranges, and the side line tells you how many weeks had those costs. The tall bars mean those costs happened more often.

🎯 Exam Tip: When given frequency tables with continuous data, always draw a histogram with contiguous bars. Remember to clearly label the x-axis with the class intervals and the y-axis with the frequencies.

 

Question 3. Draw a histogram for daily earning of 20 drug stores given in the following data :

Answer: To create the histogram, we set the daily earnings in Rupees on the x-axis and the number of drug stores on the y-axis. Each bar will represent an earning range, and its height will show how many stores fall into that specific range. This helps to visualize how the earnings are distributed among the drug stores. Here is the histogram:

In simple words: This graph shows how many shops earned money within certain ranges each day. The bottom axis shows earning amounts, and the side axis shows how many shops made that much. It helps us see which earning levels were most common.

🎯 Exam Tip: When drawing histograms from data with class intervals, ensure the bars are drawn at the correct interval boundaries. Pay attention to whether the intervals are inclusive or exclusive; for a histogram, it's usually best to use exclusive boundaries.

 

Question 4. Draw histograms for the following distributions :

(a)

Answer: For this distribution, we plot 'Marks' on the x-axis and 'Number of boys' on the y-axis. The histogram uses bars where each bar's width is the mark interval and its height is the count of boys who scored in that range. This helps us see how marks are distributed among the boys. The histogram is shown below:

In simple words: This graph shows how many boys got marks in different ranges. The bottom line shows the mark ranges, and the side line shows how many boys scored in those ranges. Taller bars mean more boys got marks in that specific range.

🎯 Exam Tip: For mark distributions, ensure that the marks on the x-axis start from the lowest value given, typically zero, and extend to cover the highest mark interval. Remember, marks are continuous data, so bars should touch.

 

(b)

Answer: We construct the histogram by plotting the 'Money earned (in Rs.)' on the x-axis and the 'Number of students' on the y-axis. Each bar in the histogram shows an earning range, and its height indicates how many students earned money within that particular range. This helps to easily understand the distribution of earnings among the students. The histogram is displayed below:

In simple words: This graph shows how many students earned different amounts of money. The bottom axis represents the money earned, and the side axis shows the number of students. Each bar's height tells us how many students fall into that earning range.

🎯 Exam Tip: Always make sure to define the range of values for both axes, especially when drawing the histogram bars. For wide ranges, consider using an appropriate scale to fit all data points clearly.

 

(c)

Answer: Since the given class intervals are inclusive (e.g., 1-10, 11-20), we first need to convert them to exclusive intervals to draw a proper histogram. We do this by adjusting the boundaries, so the upper limit of one class becomes the lower limit of the next. For example, 1-10 becomes 0.5-10.5, and 11-20 becomes 10.5-20.5. After converting, we represent these new class intervals on the x-axis and the frequencies on the y-axis to create the histogram. The conversion to exclusive form is shown in the table below:

Arranging in exclusive form.

We represent class along x-axis and frequencies along y-axis and prepare the histogram as shown here.

In simple words: When class ranges have gaps, like 1-10 then 11-20, you need to change them so they touch, like 0.5-10.5 then 10.5-20.5. After doing this, you can draw the bars for the histogram.

🎯 Exam Tip: Always check if class intervals are inclusive or exclusive. For histograms, convert inclusive intervals (e.g., 1-10, 11-20) to exclusive (0.5-10.5, 10.5-20.5) to ensure bars are continuous and touch each other correctly.

 

(d)

Answer: For this histogram, we place the class intervals along the x-axis and the frequencies along the y-axis. Notice that the class intervals are not of equal width (e.g., 10-20 is 10 units, but 50-70 is 20 units, and 70-100 is 30 units). To draw a correct histogram with unequal class widths, the heights of the bars (frequencies) must be adjusted to frequency densities. We divide each frequency by its class width to get the adjusted height. This ensures that the area of each bar is proportional to its frequency. The histogram, with adjusted heights for uneven intervals, is shown below:

In simple words: When the class groups are not all the same size, you need to draw the bars carefully. Make sure the height of each bar shows how common that group is, and the width shows the size of the group. If class sizes are unequal, you need to adjust bar heights by dividing frequency by class width.

🎯 Exam Tip: For histograms with unequal class intervals, calculate the frequency density (frequency / class width) for each interval and plot these densities on the y-axis. This ensures the area of each bar correctly represents its frequency.

 

Question 5. Given below are the marks obtained by 40 students in an examination :
29 45 23 30 40 11 48 01 15 35 40 03 12 48 49 18 30 24 25 29 31 32 25 22 27 41 12 13 02 44 07 43 09 49 19 13 32 39 25 03
Taking class-intervals 1 – 10, 11 – 20, ......., 41 - 50, make a frequency table for the above distribution and draw a histogram to represent it.

Answer: First, we organize the given marks into a frequency distribution table using the specified class intervals (1-10, 11-20, etc.). Since these are inclusive intervals, we also determine the exclusive class intervals (e.g., 0.5-10.5, 10.5-20.5) to ensure continuous bars for the histogram. Tally marks help us count the frequency for each interval. Finally, we draw the histogram with exclusive class intervals on the x-axis and frequency on the y-axis. This process creates a clear visual summary of the students' marks.

We prepare frequency distribution table as given below :

We represent class intervals (exclusive) along x-axis and frequency along y-axis and prepare the histogram to represent the above data as given below :

In simple words: First, you count how many marks fall into each given group and make a table. Since the mark groups have small gaps, you adjust them so they touch, like from 1-10 to 0.5-10.5. Then, you draw bars for these new groups.

🎯 Exam Tip: When given raw data, carefully sort it and count frequencies for each class interval. Always convert inclusive class intervals to exclusive ones before drawing a histogram to ensure the bars are adjacent.

 

Question 6. Present in the form of a frequency table the marks obtained by 50 candidates. Take the class-intervals as 11 – 20; 21 – 30... etc.
35 56 25 40 38 48 58 43 30 47 46 45 31 45 56 39 46 47 23 40 48 50 36 56 35 43 59 40 48 35 53 57 33 50 23 46 49 57 35 43 64 40 50 56 36 19 49 52 51 42

Answer: To solve this, we first find the smallest and largest marks from the list, which are 19 and 64 respectively. Then, we create a frequency distribution table with the given inclusive class intervals (11-20, 21-30, etc.) and convert them to exclusive intervals (10.5-20.5, 20.5-30.5, etc.) for accurate graphing. We count the tally marks for each interval to get the frequency, and then draw a histogram using the exclusive class intervals on the x-axis and frequencies on the y-axis. This gives a clear picture of how the marks are spread among the candidates.

Lowest data = 19,
Highest data = 64
We prepare frequency distribution table of the given data as given below :

Now we represent exclusive class intervals along x-axis and frequency along y-axis and prepare a histogram to represent the given data as shown below:

In simple words: First, find the smallest and largest marks. Then, make a table to count how many marks fall into each group (like 11-20, 21-30). Convert these groups to ranges that touch each other for the graph, and then draw bars for each range.

🎯 Exam Tip: When dealing with raw data, it's crucial to identify the lowest and highest values to ensure all data points are covered by the chosen class intervals. Converting to exclusive intervals is a necessary step for accurate histogram representation.

 

Question 7. Explain the methods of draw ing histogram and frequency polygon. Following table gives the marks distribution of 160 students in a certain class. From the above data draw a histogram and frequency polygon.

Answer: A histogram uses bars to show the frequency of data in different groups. The width of each bar is the class interval, and the height shows the frequency. For a frequency polygon, you plot points at the midpoint of the top of each histogram bar and then connect these points with straight lines. The given data is in a "more than" cumulative frequency format, so we first convert it into simple class intervals and their actual frequencies. This conversion is shown below:

We represent class intervals along x-axis and frequency along y-axis and prepare the histogram and frequency polygon by joining the mid-points of consecutive class intervals.

In simple words: A histogram is like bar chart for continuous data. To draw a frequency polygon, you first make a histogram. Then, you put a dot in the middle of the top of each bar. Connect these dots with straight lines, and you've made a frequency polygon.

🎯 Exam Tip: When drawing both a histogram and a frequency polygon, ensure the polygon starts and ends on the x-axis, typically at the midpoints of the imaginary classes before the first and after the last actual class.

 

Question 8. The number of match sticks in 40 boxes, on counting was found as given below :
44 41 42 43 47 50 51 49 43 42 40 42 44 45 49 42 46 49 45 49 45 47 48 43 43 44 48 43 46 50 43 52 46 49 52 51 47 43 43 45
Taking classes 40 – 41, 42 – 43, etc., construct the frequency distribution table for the above data. Draw a histogram to represent the above distribution.

Answer: First, we identify the lowest score (40) and highest score (52) from the given data. We then create a frequency distribution table using the specified class intervals (e.g., 40-41, 42-43). Because these are inclusive intervals, we convert them into exclusive class intervals (e.g., 39.5-41.5, 41.5-43.5) for proper histogram construction. We count the tally marks for each interval to get the frequency. Finally, we draw a histogram by plotting these exclusive class intervals on the x-axis and the frequencies on the y-axis, providing a clear visual representation of the matchstick distribution.

Lowest score = 40,
Highest score = 52
Now we prepare the frequency distribution table in exclusive form

Now we represent classes along x-axis and frequencies along y-axis and prepare the histogram representing the given data :

In simple words: To make this graph, find the smallest and largest numbers of matchsticks. Group them into ranges like 40-41, then change these ranges slightly so they touch. Count how many boxes fall into each group and then draw bars on the graph.

🎯 Exam Tip: When constructing a frequency table and histogram from raw data, ensure that the chosen class intervals cover the entire range of data from the minimum to the maximum value. Remember to adjust inclusive intervals to exclusive ones for the histogram.