Graphical Representation Class 10 ICSE — Bansal Chapter 23
The graphical representation class 10 icse chapter in the Bansal Mathematics book explains how statistical data can be shown through diagrams and charts. A diagram lets a reader grasp the meaning of data at once. The chapter studies three diagram types: the histogram, the frequency polygon, and the ogive, also called the cumulative frequency curve. Every section gives the rule for drawing one diagram and works through an example. Neatness and accuracy matter in every figure.
Download Graphical Representation Class 10 ICSE Mathematics Chapter 23 PDF
Download the Graphical Representation Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 6 pages, 7 sections, 9 questions.
What is inside Chapter 23
6 pages · 7 sections · 5 worked examples · 9 exercise questions
| Section | Title |
|---|---|
| 23.1 | Graphical Representation |
| 23.2 | Histogram |
| 23.3 | Histogram for Continuous Grouped Data |
| 23.4 | For Discontinuous Grouped Data |
| 23.5 | When Class Marks are Given |
| 23.6 | Cumulative Frequency and Cumulative Frequency Table |
| 23.7 | Cumulative Frequency Curve or an Ogive |
- EXERCISE 23: 9 questions
23.1 Graphical Representation
Statistical data can be shown as a diagram or a chart. A visual form lets the reader grasp the significance of the data at once. The diagrams must be drawn neatly and accurately, because a careless figure can mislead the eye.
Of the many diagram types in use, this ICSE Class 10 chapter studies three:
- Histogram
- Frequency polygon
- Ogive (cumulative frequency curve)
All three diagrams start from a frequency table. Students can verify the Class 10 examination requirements for statistics and graphical representation on the official CISCE website. The rest of the Class 10 ICSE Maths chapters reuse these drawing skills, so the rules in this chapter carry forward.
23.2 Histogram
A histogram is a two-dimensional graphical representation of a continuous frequency distribution. It is drawn only for continuous data. If the data is grouped in inclusive form, it must first be made continuous, which is why section 23.3 begins with a conversion step.
The rule for the rectangles is precise. The base of each rectangle is proportional to the class interval it represents. The height of each rectangle is proportional to the frequency of that class.
Because base and height each carry information, the histogram is two-dimensional. A wider class interval gives a wider base. A larger frequency gives a taller rectangle. Both proportions must be respected for the shape to be accurate.
23.3 Histogram for Continuous Grouped Data
Drawing a histogram from continuous grouped data follows three steps. First, if the data is in inclusive form, convert it to exclusive form. Second, choose suitable scales, marking class intervals on the x-axis and frequencies on the y-axis. The two scales need not be the same.
Third, construct rectangles with the class intervals as bases and the corresponding frequencies as heights. The heights follow the chosen y-scale and the bases follow the chosen x-scale. The rectangles are then placed side by side against the class boundaries.
There is one special drawing rule. When the first class starts above zero, the x-axis scale must begin at that first value. A kink or zig-zag curve is drawn near the origin to show that the scale does not begin at zero.
As a pattern, suppose a school records homework time in minutes. Intervals of 0–15, 15–30 and 30–45 with frequencies 8, 14 and 10 give rectangles of base 15 units and heights 8, 14 and 10 units. The tallest rectangle sits over the 15–30 class.
23.4 For Discontinuous Grouped Data
Some tables give class intervals in inclusive form, such as 21–29, 30–38 and 39–47. Each value belongs to one class, but the gap between the classes breaks continuity. A histogram needs continuous intervals, so the data must be converted to exclusive form.
The conversion uses the adjustment factor.
\frac{1}{2}(\text{upper limit of a class} - \text{lower limit of the next class})
For the classes 21–29 and 30–38, the factor is \frac{1}{2}(30 - 29) = 0.5. Subtract the factor from every lower limit and add it to every upper limit. The new intervals are 20.5–29.5 and 29.5–38.5, which join without a gap.
A common mistake is to adjust only one boundary. Adjusting just the upper limit of one class, or just the lower limit of the next, leaves a gap or an overlap. Apply the same factor to all lower limits and all upper limits.
23.5 When Class Marks are Given
Sometimes the table gives only the class mark, not the class intervals. The class mark is the middle value of a class. In that situation, find the class intervals first, then draw the histogram.
Consecutive class marks usually differ by a fixed amount. Half of that difference is the distance from the class mark to either end of its interval. Subtract this half-difference from each class mark to get the lower limit. Add it to each class mark to get the upper limit.
For example, class marks 15, 25, 35 and 45 have a common difference of 10. Half the difference is \frac{10}{2} = 5. The intervals become 10–20, 20–30, 30–40 and 40–50. The frequencies are then placed against these intervals and the histogram is drawn as usual.
Check that consecutive class marks are equally spaced before applying the rule. If the gap changes between classes, each interval must be built from its own two neighbouring marks.
23.6 Cumulative Frequency and Cumulative Frequency Table
The cumulative frequency of a class interval is the sum of the frequencies of all classes up to and including that interval. It answers the question: how many observations lie at or below this point?
The cumulative frequency table is built by adding one frequency at a time. Start with the first class frequency. Add the second class frequency to it, then the third, and so on. Each running total is written against its class.
For example, frequencies 5, 7, 12 and 6 give cumulative frequencies 5, 12, 24 and 30. The final total equals the sum of all frequencies.
The table also appears in a ‘less than’ form. The cumulative frequency is recorded at each upper limit. For classes 20–25, 25–30 and 30–35 with frequencies 5, 7 and 12, the less-than values are 0 at 20, 5 at 25, 12 at 30 and 24 at 35. This less-than table feeds the ogive in section 23.7.
23.7 Cumulative Frequency Curve or an Ogive
An ogive, or cumulative frequency curve, is drawn by plotting points and joining them with a free hand curve. Each point uses the upper limit of a class as its x-coordinate and the corresponding cumulative frequency as its y-coordinate.
The drawing method has five steps:
- Construct a cumulative frequency table.
- Mark the actual class limits along the x-axis.
- Mark the cumulative frequencies along the y-axis.
- Plot the point of cumulative frequency at each upper limit.
- Join the plotted points with a free hand curve.
Three properties of the ogive matter. It starts at the lower limit of the first class and ends at the upper limit of the last class. The ogive studied here is a less than ogive. Such an ogive is always a rising curve.
As a pattern, marks 10–20 and 20–30 with frequencies 6 and 9 give cumulative frequencies 6 and 15. The plotted points are (20, 6) and (30, 15). Joining them with the remaining points gives the curve.
Graph questions in this book often fix a scale in advance, such as 2\mathrm{cm} = \text{₹}500 on the x-axis and 2\mathrm{cm} = 10 workers on the y-axis. The origin may start above zero, as at 6500. Students who want more practice with cumulative tables can move on to measures of central tendency, which uses the same tables.
What to practise from Graphical Representation Class 10 ICSE
EXERCISE 23 holds 9 questions. The set covers drawing histograms from frequency distributions, converting inclusive intervals into exclusive form, recovering class intervals from class marks, plotting ogives, and constructing frequency tables from given lists of numbers.
The exercise includes graph paper questions with fixed scales. One question uses 2\mathrm{cm} = \text{₹}500, starts the origin at ₹6500 on the x-axis, and uses 2\mathrm{cm} = 10 workers on the y-axis. Practise these with a sharp pencil and a ruler so the plotted points sit exactly on the grid.
Graphical Representation Class 10 ICSE frequently asked questions
What does graphical representation class 10 icse cover in this chapter?
This chapter in the Class 10 ICSE Maths book covers three diagram types: the histogram, the frequency polygon, and the ogive. It explains the drawing rule for each, the conversion of inclusive intervals to exclusive form, and the steps for a cumulative frequency curve.
How do I convert inclusive class intervals into exclusive form for a histogram?
Find the adjustment factor: half the difference between the upper limit of a class and the lower limit of the next class. Subtract this factor from every lower limit and add it to every upper limit.
How do I draw a histogram when only class marks are given?
Find the class intervals first. Take half the difference between consecutive class marks, subtract it from each mark to get the lower limit, and add it to get the upper limit. Then draw the histogram from the recovered intervals.
How do I plot a cumulative frequency curve or ogive in Class 10 ICSE?
Build a cumulative frequency table first. Then plot each cumulative frequency against the upper limit of its class and join the points with a free hand curve. The curve starts at the lower limit of the first class and ends at the upper limit of the last class.
Why does a less than ogive always rise?
A less than ogive plots cumulative frequencies that only grow as you move to higher classes. Each new upper limit adds more observations, so the y-coordinate always increases and the curve always rises.
The syllabus this chapter follows is set by the CISCE board.
Reference: ICSE Class 10 Mathematics textbook material, chapter 23: Graphical Representation — CISCE-aligned.
More from this book
Explore Class 10 Maths Notes
Related chapters: