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

S Chand Class 10 ICSE Maths Solutions Chapter 19 Histogram and Ogive Ex 19(b)

 

Question 1.

Answer: First, we plot the class intervals on the x-axis and the frequencies on the y-axis to draw the histogram. After that, we locate the tallest rectangle in the histogram. Inside this tallest rectangle, we draw two diagonal lines: one from the top-left corner (A) to the top-right corner of the next bar (C), and another from the top-left corner of the previous bar (B) to the top-right corner of the current bar (D). These two lines will cross each other at a point, which we call P. From point P, we draw a straight line downwards to the x-axis. This line meets the x-axis at point Q. The value at point Q on the x-axis represents the mode, which is approximately 15 in this case. Drawing a histogram helps visualize the distribution of data.
In simple words: We draw a bar graph (histogram) with the given data. Then, we find the tallest bar and draw crossed lines within it. Where these lines meet, we drop a line down to the bottom axis. That point on the bottom axis is the mode, which is about 15.

🎯 Exam Tip: When drawing a histogram for grouped data, remember to choose an appropriate scale for both axes to ensure clarity. The mode is always found within the tallest bar.

 

Question 2.

Answer: First, we need to change the given class intervals into their exclusive form. This means adjusting the limits so that there are no gaps between classes. We subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class.Next, we draw the histogram using these adjusted class intervals on the x-axis and the frequencies (f) on the y-axis. Once the histogram is drawn, we identify the tallest rectangle, which represents the modal class. Inside this tallest rectangle, we connect the top-left corner (A) to the top-right corner of the next bar (C), and the top-left corner of the previous bar (B) to the top-right corner of the current bar (D). These lines cross at point P. From P, we draw a perpendicular line down to the x-axis, meeting it at Q. The value at Q on the x-axis is the mode, which is approximately 18.8, rounded to 19. The mode shows the most frequent value in the data set.
In simple words: First, we change the mark ranges so they don't have gaps. Then we draw a bar graph (histogram). In the highest bar, we draw crossed lines. Where they cross, we draw a line down to the marks axis. This point shows the most common mark, which is about 19.

🎯 Exam Tip: Always convert inclusive class intervals to exclusive class intervals before drawing a histogram, especially when finding the mode, to avoid gaps and ensure continuous bars.

 

Question 3.

Answer: We begin by converting the given inclusive class intervals into exclusive ones to make sure there are no gaps between the data ranges. To do this, we subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class.Next, we plot these exclusive class intervals on the x-axis and their frequencies (f) on the y-axis to create a histogram. After drawing the bars, we find the tallest rectangle, which shows the modal class. Inside this tallest bar, we draw two diagonal lines: one from the top-left corner (A) to the top-right corner of the next bar (C), and another from the top-left corner of the previous bar (B) to the top-right corner of the current bar (D). These lines meet at point P. From point P, we drop a vertical line down to the x-axis, which intersects at point Q. This point Q gives us the mode, which is approximately 29. Understanding the mode helps us know the most common value.
In simple words: First, we adjust the mark ranges to remove gaps. Then, we make a bar graph (histogram) using these adjusted ranges. We find the tallest bar and draw crossed lines inside it. Where they cross, we draw a line down to the marks axis. This point on the axis tells us the mode, which is about 29.

🎯 Exam Tip: When drawing lines B-D and A-C to find the mode, ensure that line B-D starts from the top-left of the modal class's bar and connects to the top-right of the previous bar. If there's no previous bar, it starts from the origin on the x-axis, just below the modal class's lower limit.

 

Question 4.

Answer: Since the mid-values are given, we first need to find the class intervals. We can see that the difference between consecutive mid-values is 4 (e.g., 64 - 60 = 4). So, the class width is 4. To get the lower limit of a class, we subtract half the class width (4/2 = 2) from the mid-value. To get the upper limit, we add half the class width to the mid-value.
So, for the mid-value 60: Lower limit = \( 60 - 2 = 58 \) Upper limit = \( 60 + 2 = 62 \) The first class is 58-62. We apply this rule to all mid-values to form the classes:Next, we draw a histogram with these classes on the x-axis and their frequencies on the y-axis. Inside the tallest rectangle (modal class), we draw lines A-C and B-D which cross at point P. From P, we draw a line perpendicular to the x-axis, meeting it at Q. The value at Q on the x-axis is the mode, which is approximately 60.4, rounded to 60. The process helps us find the most common range.
In simple words: First, we use the middle numbers to find the actual ranges for each group. Then, we draw a bar graph (histogram). In the tallest bar, we draw crossed lines. Where they meet, we drop a line to the bottom axis. This point shows the mode, which is about 60.

🎯 Exam Tip: When given mid-values, remember to calculate the class width by finding the difference between two consecutive mid-values. Then, halve this width to find the adjustment value for determining the class limits.

 

Question 5. Draw a histogram from the following data and measure the modal value:
(a)

(b)Answer:
(a) We plot the size classes on the x-axis and their frequencies on the y-axis to draw the histogram. The tallest bar represents the modal class. Inside this tallest bar, we draw two diagonal lines: one from its top-left corner to the top-right corner of the next bar, and the other from the top-left corner of the previous bar to its top-right corner. These lines cross at point P. From P, we draw a perpendicular line down to the x-axis, which meets at point Q. The value at Q is the mode, which is approximately 33. This method helps us visually find the most frequent value.
In simple words: We make a bar graph with the given data. In the tallest bar, we draw crossed lines. Where they meet, we drop a line to the bottom axis. This point gives us the mode, which is about 33.

🎯 Exam Tip: When the modal class is not the first or last bar, ensure that the lines B-D and A-C are correctly drawn from the top corners of the adjacent bars to the top corners of the modal bar.

 

(b) We draw a histogram by placing the size classes on the x-axis and their frequencies on the y-axis. Then, we find the tallest rectangle, which is the modal class. Inside this tallest rectangle, we draw a diagonal line from its top-left corner (A) to the top-right corner of the next bar (C), and another line from the top-left corner of the previous bar (B) to its top-right corner (D). These lines cross at point P. From P, we drop a vertical line to the x-axis, meeting it at Q. The value at Q on the x-axis gives us the mode, which is approximately 68. This visual method helps quickly identify the central tendency.
In simple words: We draw a bar graph for the given data. We find the tallest bar and draw crossed lines inside it. Where they cross, we draw a line down to the bottom axis. This point shows the mode, which is about 68.

🎯 Exam Tip: When the first bar is the modal class, the line B-D (from the previous bar's top-right to the modal bar's top-right) should originate from the point on the x-axis that marks the lower limit of the modal class, assuming zero frequency for the non-existent previous class.