Graphical Representation Class 10 ICSE — A. Das Gupta Ch 27
Graphical representation class 10 ICSE covers the chapter on Graphical Representation of Statistical Data from A. Das Gupta’s Mathematics. This ICSE chapter explains why graphs make data more noticeable, easily intelligible, and easy to compare. You will learn three graph types: the histogram, the frequency polygon, and the cumulative frequency curve or ogive. Each has its own construction method, and Exercise 27 tests all three.
Download Graphical Representation of Statistical Data Class 10 ICSE Mathematics Chapter 27 PDF
Download the Graphical Representation of Statistical Data Class 10 ICSE Mathematics chapter PDF from the A. Das Gupta textbook to revise offline or print for practice. It is the full chapter as printed in the book — 12 pages, 5 sections, 13 questions.
What is inside Chapter 27
12 pages · 5 sections · 1 worked examples · 13 exercise questions
| Section | Title |
|---|---|
| 1 | Graphical Representation of Data |
| 2 | (A) HISTOGRAM |
| 3 | Method of Drawing a Histogram |
| 4 | (B) FREQUENCY POLYGON |
| 5 | Method of Drawing A Frequency Polygon |
- EXERCISE 27: 13 questions
1 Graphical Representation of Data
Tabular representation of data is systematic, but the chapter opens with a key point: when numbers are drawn as pictures or graphs, they become more noticeable and easily intelligible. Comparisons also become easier. A table forces you to read every row. A graph lets you see the shape of the data at a glance. That is why statistical data is so often presented graphically in the Class 10 ICSE syllabus.
This Class 10 ICSE chapter deals with three graph types in order. First is the histogram, built from rectangles with no gaps between them. Second is the frequency polygon, formed by joining plotted points. Third is the cumulative frequency curve, also called an ogive. Each has its own definition and its own drawing method. The methods build on one another, so the order matters.
The three graphs are connected. A histogram and a frequency polygon are often drawn on the same axes. An ogive starts from the same frequency table. Learn the steps for each one and you will handle Exercise 27’s thirteen questions without confusion. For more statistical tools, browse the Class 10 ICSE Mathematics chapter index.
2 (A) HISTOGRAM
A histogram is a graphical representation of a frequency distribution in exclusive form. Exclusive form means continuous class intervals, such as 10–20, 20–30, with no overlap and no gap between them. The histogram draws rectangles. Each class interval forms the base of a rectangle, and the corresponding frequency forms its height.
The defining rule is that there is no gap between any two successive rectangles. This single rule separates a histogram from a bar diagram. A bar diagram has separated bars for separate categories. A histogram shows continuous data, so the rectangles must touch one another.
Students often draw small gaps between rectangles out of habit. That error changes the figure into a bar diagram, and the exam will mark it wrong. If your data is in inclusive form, you must convert it to exclusive form before drawing. That conversion is the first step of the method in the next section.
You should be able to reproduce this definition in your own words in an exam. It is the definition the entire method rests on.
3 Method of Drawing a Histogram
The method has three steps in order.
- If the distribution is in inclusive form, convert it into exclusive form.
- Choose suitable scales. Mark class intervals on the x-axis and frequencies on the y-axis. The two scales need not be the same.
- Construct rectangles with class intervals as bases and frequencies as heights.
The conversion in step 1 uses half the gap between consecutive intervals. Take intervals 7–14, 15–22, 23–30 with frequencies 5, 12, 8. The gap between 14 and 15 is 1, so half the gap is 0.5. Subtract 0.5 from each lower limit and add 0.5 to each upper limit. You get 6.5–14.5, 14.5–22.5, 22.5–30.5. The intervals are now continuous and exclusive.
Sometimes you are given class marks instead of class intervals. The class size is the difference between consecutive class marks. If class marks are 150, 160, 170, then the class size is h = 160 - 150 = 10. Rebuild each interval around its class mark, so the mark 150 gives the interval 145–155. Then draw the histogram normally.
This conversion step appears often in graphical representation class 10 ICSE questions, so master it before moving to the polygon.
4 (B) FREQUENCY POLYGON
The frequency polygon uses notation you need to read carefully. Let x_1, x_2, x_3, \ldots, x_n be the class marks, meaning the mid-points of the class intervals. Let f_1, f_2, f_3, \ldots, f_n be the corresponding frequencies. A class mark is the mid-point of an interval. The interval 10–20 has class mark 15, because (10 + 20)/2 = 15.
You plot the points (x_1, f_1), (x_2, f_2), (x_3, f_3), \ldots, (x_n, f_n) on graph paper. Then you join these points with straight line segments. The result is a polygon, but it is not yet closed at the ends.
The chapter completes the figure by taking two more classes, called imagined classes, one at the beginning and one at the end of the distribution. These imagined classes have frequency zero. Their purpose is to bring the polygon down to the x-axis at both ends, so the figure closes into a proper polygon. The next section gives the full six-step method, including how to find the class marks of these imagined classes.
5 Method of Drawing A Frequency Polygon
Drawing a frequency polygon follows six steps.
- Calculate the class marks x_1, x_2, \ldots, x_n of the given class intervals.
- Mark the class marks along the x-axis on a suitable scale.
- Mark the frequencies along the y-axis on a suitable scale.
- Join the plotted points (x_1, f_1), (x_2, f_2), \ldots, (x_n, f_n) with line segments.
- Take two class intervals of frequency zero, one at the beginning and one at the end of the frequency table. Find their class marks. These are the imagined classes.
- Join the mid-point of the first class interval to the mid-point of the imagined class at the beginning. Do the same for the last class interval and the imagined class at the end.
Work through a pattern with five intervals. Marks 5–10, 10–15, 15–20, 20–25, 25–30 have frequencies 3, 8, 12, 7, 5. The class marks are 7.5, 12.5, 17.5, 22.5, 27.5. Add imagined classes 0–5 and 30–35, both with frequency zero. Their class marks are 2.5 and 32.5.
Plot the seven points (2.5, 0), (7.5, 3), (12.5, 8), (17.5, 12), (22.5, 7), (27.5, 5), (32.5, 0) on graph paper. Join them in order with line segments. The polygon now closes at both ends. You have drawn the frequency polygon for this distribution.
The methods in this graphical representation class 10 ICSE chapter build on each other. A polygon uses the same class marks you found for the histogram. Only the imagined classes are new.
C) Cumulative Frequency Curve or Ogive
An ogive represents a frequency distribution through cumulative frequencies. Mark the upper class limits along the x-axis and the corresponding cumulative frequencies along the y-axis. Join these points by a free hand curve, not straight line segments. This curve is called an ogive. The cumulative frequency at each point is the running total of all frequencies up to that class.
To complete the ogive, plot the lower limit of the first class interval on the x-axis. Join this point to the first point of the curve. This completion point has cumulative frequency zero, because no data falls below the first interval. Without this step, the curve does not start at the x-axis.
One edge case saves time. When the data is already in ‘less than’ cumulative form, the cumulative frequency table is already built. You plot the points directly, exactly as you would for any ogive. Question 11 of Exercise 27 uses this form, and students who recognise it skip the table-building step entirely. Learn to recognise this form at a glance, because it appears in board questions.
Method of Drawing an Ogive
An ogive needs a running total before any point can be plotted. That is why the cumulative frequency table comes first. The method has five steps in order.
- If the distribution is in inclusive form, convert it into exclusive form.
- Prepare the cumulative frequency table.
- Mark the upper limits of the class intervals on the x-axis and the cumulative frequencies on the y-axis.
- Plot the lower limit of the first class interval with cumulative frequency 0.
- Join all these points by a free hand curve to obtain the required ogive.
Work through a pattern with heights. Intervals 120–130, 130–140, 140–150, 150–160, 160–170, 170–180 have frequencies 4, 9, 15, 12, 6, 4. The cumulative frequencies are 4, 13, 28, 40, 46, 50.
Plot the points (130, 4), (140, 13), (150, 28), (160, 40), (170, 46), (180, 50) on graph paper. Then add the completion point (120, 0), the lower limit of the first interval with cumulative frequency zero. Join all seven points with a free hand curve to get the ogive.
The step students often miss is the completion point. Without it, the ogive does not start at the x-axis. Always plot the lower limit of the first class interval with cumulative frequency 0 before drawing the curve.
What to practise from Graphical Representation Class 10 ICSE
Exercise 27 has 13 questions. Each one trains a specific method from this chapter. The questions group into three sets by graph type.
- Histograms (Q1–Q5): Q1 and Q2 draw a histogram from an exclusive table. Q3 needs inclusive-to-exclusive conversion first. Q5 gives class marks, so you rebuild the intervals using the class size h = 160 - 150 = 10.
- Frequency polygons (Q6–Q9): Q6 draws a polygon alone. Q7 draws a histogram and polygon on the same graph. Q9 draws the polygon on top of the histogram using the class marks.
- Ogive (Q10–Q13): Q10 builds the cumulative table from raw frequencies. Q11 is already in ‘less than’ cumulative form, so you plot directly. Q12 and Q13 need inclusive-to-exclusive conversion first.
Q2 carries a board year mark of 2005, so this exercise type has appeared in board papers. Attempt every question once you have practised each method. You can verify the syllabus structure on the official CISCE website. For more chapters from this book, check the Class 10 books index.
Graphical Representation Class 10 ICSE frequently asked questions
What is the difference between a histogram and a bar diagram in graphical representation class 10 ICSE?
A histogram has no gaps between rectangles because it shows continuous data in exclusive form. A bar diagram has separated bars for separate categories. Drawing gaps between histogram rectangles turns the figure into a bar diagram, which is the most common mistake in this chapter.
How do I convert inclusive class intervals to exclusive form before drawing a histogram?
Find the gap between the upper limit of one interval and the lower limit of the next. Take half of that gap. Subtract it from every lower limit and add it to every upper limit. For intervals 7–14 and 15–22, the gap is 1, so you subtract and add 0.5 to get 6.5–14.5 and 14.5–22.5.
What are imagined classes in a frequency polygon?
Imagined classes are two extra classes added at the beginning and end of the distribution, each with frequency zero. Their class marks give the end points of the polygon. Joining these points brings the polygon down to the x-axis and closes the figure.
How do I draw an ogive when the data is already given as less than cumulative frequencies?
Skip the cumulative frequency table step, because the table is already built. Plot the upper limits against the given cumulative frequencies directly. Then add the lower limit of the first interval with cumulative frequency zero, and join all points with a free hand curve.
How do I find the class mark of a class interval?
Add the lower limit and the upper limit of the interval, then divide by two. For the interval 10–20, the class mark is (10 + 20)/2 = 15. This is the mid-point of the interval, and it is the x-coordinate used when plotting a frequency polygon.
Reference: ICSE Class 10 Mathematics textbook material, chapter 27: Graphical Representation of Statistical Data — CISCE-aligned.
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