Graphical Representation

Vardaan Learning Institute • ICSE Class 10 Mathematics • Chapter 23

1. Overview of Graphical Methods

Histogram: A graphical representation of a continuous frequency distribution using adjacent vertical rectangles.

Frequency Polygon: A curve formed by joining the mid-points (class marks) of the tops of the rectangles of a histogram with straight lines.

Ogive (Cumulative Frequency Curve): A smooth freehand curve obtained by plotting cumulative frequencies against class limits.

2. Histograms & Mode Estimation

Steps to Estimate Mode Graphically using Histogram:

Draw the histogram for the continuous frequency distribution.
Locate the highest rectangle (Modal Class).
From the top-left corner of the highest rectangle, draw a line segment to the top-left corner of the adjacent rectangle on the right.
From the top-right corner of the highest rectangle, draw a line segment to the top-right corner of the adjacent rectangle on the left.
Draw a vertical line from the point of intersection of these two line segments to the x-axis.
The value on the x-axis corresponds to the Mode.

3. Ogive & Quartile Estimations

Less Than Ogive Plotting:

Plot Upper Class Limits on x-axis and Cumulative Frequency (c.f.) on y-axis.

Locating Measures on Ogive ($N = \text{Total Frequency}$):

Median ($M$): Locate $\frac{N}{2}$ on y-axis, draw horizontal line to Ogive, drop perpendicular to x-axis.
Lower Quartile ($Q_1$): Locate $\frac{N}{4}$ on y-axis, drop perpendicular to x-axis.
Upper Quartile ($Q_3$): Locate $\frac{3N}{4}$ on y-axis, drop perpendicular to x-axis.
Interquartile Range: $$Q_3 - Q_1$$
Semi-Interquartile Range: $$\frac{Q_3 - Q_1}{2}$$

4. Solved Master Example

Problem: Given the distribution of marks of 50 students:

Marks	0-10	10-20	20-30	30-40	40-50
No. of Students ($f$)	5	12	20	9	4
c.f.	5	17	37	46	50

Find: Median, $Q_1$, $Q_3$, and number of students scoring above 35 marks.

Solution:

1. Total $N = 50$.

2. $\frac{N}{2} = 25 \implies \text{Median mark from Ogive} \approx 24$.

3. $\frac{N}{4} = 12.5 \implies Q_1 \approx 16$.

4. $\frac{3N}{4} = 37.5 \implies Q_3 \approx 30$.

Exam Practice Questions (ICSE PYQs)

PYQ 2023

BOARD Draw an Ogive for the following distribution and estimate: (i) Median (ii) Interquartile range (iii) Number of students scoring more than 75% marks.