x̄ = Σfᵢxᵢ / Σfᵢ Histogram: no gaps (continuous) Bar graph: gaps (discrete) Class mark = (upper+lower)/2
x̄
Chapter 12  ·  Class IX Mathematics

Collecting, Organising, and Interpreting Data

Statistics

From Raw Numbers to Meaningful Insights — The Science of Data in Class IX

📖 Read Full Notes ⚡ Formula Capsules

Chapter Snapshot

8Concepts
6Formulae
8–10%Exam Weight
4–5Avg Q's
Easy-ModerateDifficulty

Why This Chapter Matters for Exams

CBSE Class IXNTSEState Boards

Statistics contributes 8–10 marks in CBSE Class IX Boards, primarily through graph drawing (bar graphs, histograms, frequency polygons) and calculation of mean, median, and mode. Histogram vs bar graph distinction is a guaranteed 1-mark question. Mean for ungrouped and grouped data appears as a 3-mark question. NTSE includes data interpretation and statistical reasoning questions.

Key Concept Highlights

Data Collection: Primary and Secondary
Presentation of Data: Frequency Distribution Table
Ungrouped and Grouped Frequency Distribution
Bar Graph
Histogram
Frequency Polygon
Measures of Central Tendency: Mean
Measures of Central Tendency: Median
Measures of Central Tendency: Mode

Important Formula Capsules

$\bar{x} = \frac{\sum x_i}{n}\ (\text{mean of ungrouped data})$
$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\ (\text{mean of grouped data})$
$\text{Median (odd }n\text{): middle value when arranged in order}$
$\text{Median (even }n\text{): average of two middle values}$
$\text{Mode: value occurring most frequently}$
$\text{Class mark} = \frac{\text{upper limit} + \text{lower limit}}{2}$

What You Will Learn

Navigate to Chapter Resources

📖 Full Notes Complete NCERT Coverage 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank CBSE · NTSE · State Boards 📘✔️ Exercises Step-by-step NCERT Solutions ✔️❌ True / False Concept-based True & False

🏆 Exam Strategy & Preparation Tips

Graph questions in CBSE require ruler-drawn, neatly labelled diagrams with a scale — always show the scale and axes titles. The key distinction: histogram has no gaps between bars (continuous data); bar graph has gaps (discrete/categorical data). For mean with grouped data, always make a table with columns for xᵢ, fᵢ, and fᵢxᵢ before applying the formula. Time investment: 2–3 days.

Chapter 12 · CBSE · Class IX
📊

Graphical Representation of Data

NCERT Class 9 Mathematics Statistics Data Handling Bar Graphs Histograms Frequency Polygons Data Representation Graphical Analysis
📋
Table of Contents
15 sections
🗺️ Overview

Graphical Representation of Data is the process of presenting numerical information in a visual form so that it becomes easier to understand, compare, interpret, and analyze. Instead of studying long tables of numbers, graphs help us observe patterns, trends, distributions, and relationships at a glance.

In everyday life, graphical representation is widely used in newspapers, business reports, weather forecasts, scientific studies, sports statistics, economics, population surveys, and educational research. A properly constructed graph can communicate complex information more effectively than several pages of numerical data.

📘 Definition
🗒️ Importance
Importance of Graphical Representation
  • Transforms complex numerical data into an easy-to-understand visual form.
  • Facilitates quick comparison among different categories.
  • Helps identify trends, patterns, and relationships.
  • Reduces the effort required to interpret large data sets.
  • Makes reports and presentations more effective.
  • Provides a clear picture of maximum, minimum, and average tendencies.
🛠️ Real-Life Applications
  • Population census and demographic studies.
  • Weather forecasting and climate analysis.
  • Business and sales performance reports.
  • Sports statistics and tournament analysis.
  • Scientific experiments and research.
  • School examination performance analysis.
  • Government surveys and economic planning.
📌 Note
🎨 SVG Diagram
Axes of a Graph
X-axis Y-axis O
📌 Types of Graphical Representation Studied in Class 9
📘 Definition
Bar Graph
📘 Definition
Histogram
📘 Definition
Frequency Polygon
🔄 General Steps for Drawing a Graph
  • 1
    Study the data carefully
  • 2
    Select an appropriate graph type.
  • 3
    Choose a suitable scale.
  • 4
    Draw X-axis and Y-axis.
  • 5
    Label both axes clearly.
  • 6
    Plot values accurately.
  • 7
    Write a suitable title.
  • 8
    Mention the scale used.
✏️ Example
Solved Example

The marks obtained by four students are:

Student A B C D
Marks 40 55 70 50
Since separate categories are being compared, a bar graph is the most suitable graphical representation.
  1. 1

    Draw the X-axis and Y-axis.

  2. 2

    Represent students on the X-axis.

  3. 3

    Represent marks on the Y-axis.

  4. 4

    Choose a suitable scale.

  5. 5

    Draw bars corresponding to the marks.

Student Marks Bar Chart Data visualization for student marks with Deep Sea Blue background and visible axes Student A: 40 A 40 Student B: 55 B 55 Student C: 70 C 70 Student D: 50 D 50 Student Marks Comparison

Student C scored the highest marks while Student A scored the lowest marks.

⚡ Exam Tip
❌ Common Mistakes
  • Using an unsuitable scale.
  • Forgetting to write graph title.
  • Incorrect labeling of axes.
  • Leaving unequal gaps between bars.
  • Leaving spaces between histogram rectangles.
  • Plotting incorrect frequencies.
  • Ignoring units.
📋 Case Study

A school recorded participation in various co-curricular activities:

Activity Students
Sports 60
Music 45
Dance 50
Art 35

Questions

  1. Which activity has maximum participation?
  2. Which activity has minimum participation?
  3. Which graph is most suitable?
  4. Find the difference between maximum and minimum participation.

Solution

  1. Sports
  2. Art
  3. Bar Graph
  4. \[60 - 35 = 25\]
⚡ Quick Revision
  • Graphs visually represent data.
  • Bar Graph is used for discrete data.
  • Histogram is used for continuous data.
  • Frequency Polygon shows trends in distribution.
  • Proper scale and labeling are essential.
  • Graphs simplify interpretation and comparison.
📊

Bar Graphs

📋
Table of Contents
13 sections
🗺️ Overview

A bar graph is one of the simplest and most widely used methods of graphical representation of data. It uses rectangular bars of equal width to represent different categories, groups, or classes. The length or height of each bar is proportional to the value it represents. Bar graphs provide an easy visual comparison between different categories and help us understand data quickly without examining lengthy tables.

Bar graphs are extensively used in statistics, economics, business reports, population studies, sports analysis, educational surveys, and scientific investigations. Since human eyes can compare lengths more easily than numerical values, bar graphs make data interpretation faster and more effective.

📘 Definition
🔷 Characteristics of a Bar Graph
🔷 Characteristics
  • Bars are rectangular in shape.
  • All bars have equal width.
  • Bars are separated by equal gaps.
  • The height or length of a bar represents the value of the category.
  • Bars may be drawn vertically or horizontally.
  • Suitable for discrete and categorical data.
  • Provides quick visual comparison among categories.
📌 Parts of a Bar Graph
🎨 SVG Diagram
Structure of a Bar Graph
Performance Comparison Analysis A 24 B 34 C 44 D 30 Students Marks 10 20 30 40 50
🗂️ Types of Bar Graphs
Vertical Bar Graph
In a vertical bar graph, bars are drawn vertically upward from the horizontal axis. The height of each bar represents the value of the corresponding category.
Horizontal Bar Graph
In a horizontal bar graph, bars are drawn horizontally. This type is useful when category names are lengthy and difficult to fit along the horizontal axis.
Double Bar Graph
A double bar graph is used to compare two related sets of data simultaneously.
Example: Comparing marks obtained by boys and girls in different subjects.
Multiple Bar Graph
Multiple bar graphs compare more than two related data sets using groups of bars.
✅ Advantages of Bar Graphs
  • Easy to draw and understand.
  • Provides instant comparison.
  • Suitable for large data sets.
  • Visually attractive presentation.
  • Helps identify highest and lowest values quickly.
  • Useful in reports, surveys, and presentations.
⚠️ Limitations
Limitations of Bar Graphs
  • Cannot show detailed distribution of continuous data.
  • Less suitable when there are too many categories.
  • Exact values may not always be easily determined from visual inspection.
🔄 Steps to Draw a Bar Graph
  • 1
    Study the data carefully.
  • 2
    Select a suitable scale.
  • 3
    Draw X-axis and Y-axis.
  • 4
    Mark categories on one axis.
  • 5
    Mark values on the other axis.
  • 6
    Draw bars of equal width.
  • 7
    Leave equal spacing between bars.
  • 8
    Write the title and scale clearly.
✏️ Example
Solved Eample
The number of books read by students during a month is given below: \[ \begin{array}{|l|c|c|c|c|} \hline \hline \text{Student} & \text{A} & \text{B} & \text{C} & \text{D} \\ \hline \text{Books Read} & 5 & 8 & 10 & 6 \\ \hline \hline \end{array} \]
Individual categories are being compared; therefore a bar graph is the most appropriate graphical representation.
  1. 1
    Take students along the X-axis.
  2. 2
    Take number of books along the Y-axis.
  3. 3
    Choose a convenient scale.
  4. 4
    Draw bars corresponding to values 5, 8, 10, and 6.
Books Read by Students (Monthly) A 5 B 8 C 10 D 6 Students Books 2 4 6 8 10
Student C read the maximum number of books, while Student A read the minimum number.
⚡ Exam Tip
❌ Common Mistakes
  • Choosing an unsuitable scale.
  • Forgetting graph title.
  • Incorrect axis labeling.
  • Leaving unequal gaps in bar graphs.
  • Leaving gaps in histograms where rectangles should touch.
  • Plotting wrong frequencies.
  • Ignoring units and class intervals.
📋 Case Study

The following data show the number of medals won by four houses in a school sports competition:

House Red Blue Green Yellow
Medals 18 25 21 15

Questions:

  1. Which house won the maximum medals?
  2. Which house won the minimum medals?
  3. How many more medals did Blue House win than Yellow House?
  4. Which graph is most suitable to represent this data?

Answers:

  1. Blue House
  2. Yellow House
  3. \[25 - 15 = 10\]
  4. Bar Graph
📝 Summary
📊

Example 1

📋
Table of Contents
7 sections
❓ Question
In a particular section of Class IX, 40 students were asked about the months of their birth and the following graph was prepared for the data so obtained:
Observe the bar graph given above and answer the following questions:
(i) How many students were born in the month of November?
(ii) In which month were the maximum number of students born?
🎨 SVG Diagram
Fig 12.1
0 2 4 6 8 Number of Students Months of Birth 3 4 2 2 5 1 2 7 3 4 4 4 Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
💡 Concept
Concept Used
🗺️ Roadmap
Solution Roadmap
  1. Locate the required category on the X-axis.
  2. Observe the corresponding bar.
  3. Read its height using the Y-axis scale.
  4. For maximum or minimum values, compare the heights of all bars.
🧩 Solution
  1. (i) How many students were born in the month of November?

    Locate the bar corresponding to November. The height of the bar reaches the value 4 on the vertical axis.

    Therefore,

    Number of students born in November = 4

  2. (ii) In which month were the maximum number of students born?

    To answer this question, compare the heights of all the bars. The tallest bar corresponds to the month of August.

    The height of this bar represents 7 students.

    Therefore,

    The maximum number of students were born in August.

    Number of students born in August = 7

👁️ Observation
Additional Observations from the Graph
⚡ Exam Tip
❌ Common Mistakes
  • Reading the height of a neighboring bar instead of the required bar.
  • Ignoring the scale mentioned on the graph.
  • Confusing category names on the horizontal axis.
  • Assuming values without checking the actual height.
🌟 Key Learning
📊

Example 2

📋
Table of Contents
9 sections
❓ Question
A family with a monthly income of ` 20,000 had planned the following expenditures per month under various heads: \[ \begin{array}{|l|c|} \hline \hline \text{Heads} & \text{Expenditure (in thousand rupees)}\\ \hline \text{Grocery} & 4\\ \hline \text{Rent} & 5\\ \hline \text{Education of children} & 5\\ \hline \text{Medicine} & 2\\ \hline \text{Fuel} & 2\\ \hline \text{Entertainment} & 1\\ \hline \text{Miscellaneous} & 1\\ \hline \end{array} \] Draw a bar graph for the data above.
💡 Concept
Concept Used
📌 Understanding the Data
🔄 Step-by-Step Construction of the Bar Graph
  • 1
    Draw two perpendicular axes.
  • 2
    Mark expenditure heads on the horizontal axis (X-axis).
  • 3
    Mark expenditure values on the vertical axis (Y-axis).
  • 4
    Since the maximum expenditure is 5 thousand rupees, choose a convenient scale: <br> \[ 1 \text{ unit} = \text{₹}1,000 \]
  • 5
    Draw bars of equal width.
  • 6
    Leave equal gaps between consecutive bars.
  • 7
    \[ \begin{array}{|l|c|} \hline \text{Draw bars having heights:}\\ \hline \hline \text{Grocery} & 4\text{ units}\\ \hline \text{Rent} & 5\text{ units}\\ \hline \text{Education of children} &\text{ 5 units}\\ \hline \text{Medicine} & 2\text{ units}\\ \hline \text{Fuel} & 2\text{ units}\\ \hline \text{Entertainment} &\text{ 1 unit}\\ \hline \text{Miscellaneous} &\text{ 1 unit}\\ \hline \end{array} \]
  • 8
    Label both axes and provide a suitable title.
🎨 SVG Diagram
Fig. 12.2
0 1 2 3 4 5 6 7 Expenditure (in thousand rupees) 4 Grocery 5 Rent 5 Education of children 2 Medicine 2 Fuel 1 Entertainment 1 Miscellaneous Heads
🔍 Analysis of the Graph

Once the graph is drawn, several conclusions can be obtained immediately without performing lengthy calculations.

  • Rent and Education have the highest expenditure.
  • Entertainment and Miscellaneous have the lowest expenditure.
  • Medicine and Fuel expenses are equal.
  • Grocery expenditure is greater than Medicine and Fuel expenditure.
  • The graph provides a quick comparison of family spending habits.
🧩 Solution
  1. Draw a bar graph taking expenditure heads on the X-axis and expenditure (in thousand rupees) on the Y-axis.
  2. Use the scale:
    \[1 \text{ unit} = \text{₹}1,000\]
  3. Draw bars of equal width corresponding to the expenditure values: 4, 5, 5, 2, 2, 1, and 1 respectively.
  4. The completed graph is shown in Fig. 12.2.
⚡ Exam Tip
❌ Common Mistakes
  • Forgetting that values are given in thousand rupees.
  • Using unequal bar widths.
  • Drawing bars without a proper scale.
  • Interchanging the X-axis and Y-axis labels.
  • Not leaving equal gaps between bars.
  • Using an unnecessarily complicated scale.
📍 Key Learning
Bar graphs are powerful tools for comparing quantities belonging to different categories. They transform numerical tables into visual representations, enabling quick analysis, better understanding, and efficient decision-making.

In this example, the family's spending pattern becomes immediately visible through the graph, making it much easier to compare various expenditure heads than by studying the numerical table alone.
📊

Histogram

📋
Table of Contents
16 sections
🗺️ Overview

A histogram is a graphical representation of a frequency distribution in which data are grouped into continuous class intervals. It consists of adjoining rectangular bars whose areas are proportional to the frequencies of the corresponding classes. Unlike a bar graph, the rectangles in a histogram touch each other because the data represent a continuous variable.

Histograms are widely used in statistics, economics, business analytics, scientific research, population studies, quality control, and educational surveys. They help us understand how data are distributed and reveal important patterns such as concentration, spread, symmetry, and variation within a dataset.

📘 Definition
🌟 Why Do We Need a Histogram?
💡 Basic Concept
🔷 Characteristics of a Histogram
🔷 Characteristics
  • Used for continuous frequency distributions.
  • Rectangles touch one another.
  • No gaps are left between adjacent bars.
  • Width of each rectangle represents class interval.
  • Height of each rectangle represents frequency.
  • Area of each rectangle is proportional to frequency.
  • Provides a visual picture of data distribution.
📊 Difference Between Histogram and Bar Graph
Histogram Bar Graph
📌 Parts of a Histogram
🎨 SVG Diagram
Structure of a Histogram
Fig. 12.3
30.5 35.5 40.5 45.5 50.5 55.5 60.5 Weights (in kg) 0 2 4 6 8 10 12 14 16 Number of Students Histogram: Weight Distribution of Students academia-aeternum.com | Deep Sea Blue Theme
🔄 Steps to Draw a Histogram
  • 1
    Write the class intervals on the horizontal axis.
  • 2
    Mark frequencies on the vertical axis.
  • 3
    Choose an appropriate scale.
  • 4
    Draw adjacent rectangles for each class interval.
  • 5
    Ensure there are no gaps between rectangles.
  • 6
    Label both axes clearly.
  • 7
    Provide a suitable title.
📌 Important Note
🛠️ Real-Life Applications of Histograms
  • Analysis of examination marks.
  • Age distribution of populations.
  • Income distribution studies.
  • Rainfall analysis.
  • Quality control in manufacturing.
  • Medical and scientific research.
  • Business and market surveys.
✏️ Example
The marks obtained by students are grouped as follows: \[ \begin{array}{|l|c|} \hline\hline\\ \text{Marks}&\text{Frequency}\\\hline 0-10& 3\\\hline 10-20&7\\\hline 20-30&12\\\hline 30-40& 9\\\hline 40-50& 4\\\hline \end{array} \]
Since the data are grouped into continuous class intervals, a histogram is used.
Distribution of Student Marks 0 3 6 9 12 0 10 20 30 40 50 Marks Obtained Frequency (Students)
👁️
Observation The class interval 20–30 has the highest frequency. Therefore, most students scored between 20 and 30 marks.
⚡ Exam Tip
❌ Common Mistakes
  • Leaving gaps between histogram bars.
  • Using categories instead of class intervals.
  • Incorrect plotting of frequencies.
  • Ignoring the scale.
  • Confusing class intervals with frequencies.
  • Not labeling axes properly.
📋 CBSE Competency-Based Question

A histogram shows the distribution of marks obtained by students in a mathematics test. The tallest rectangle corresponds to the class interval 40–50.

  1. What does the tallest rectangle indicate?
  2. Can we conclude that every student scored between 40 and 50? Explain.
  3. Which class interval is called the modal class?

Answer

  1. The class interval 40–50 has the highest frequency.
  2. No. It only means the largest number of students belong to that interval.
  3. The class interval having the highest frequency is called the modal class.
📌 Advanced Insight for Curious Learners
📝 Summary
📊

Example 3

📋
Table of Contents
11 sections
❓ Question

Draw a histogram for the following frequency distribution showing the weights of students.

Weight (in kg) Number of Students
30.5 – 35.5 9
35.5 – 40.5 6
40.5 – 45.5 15
45.5 – 50.5 3
50.5 – 55.5 1
55.5 – 60.5 2
💡 Concept
Concept Used
🤔 Why Is a Histogram Used Here?
  • Weight is a continuous variable.
  • Observations are grouped into class intervals.
  • The frequencies correspond to each class interval.
  • Histogram helps visualize the distribution of weights.
  • The highest frequency class can be identified immediately.
📌 Note
Understanding the Frequency Distribution
🗒️ Soution
  1. Draw two perpendicular axes.
  2. Represent weight intervals on the horizontal axis (X-axis).
  3. Represent the number of students on the vertical axis (Y-axis).
  4. Since the data start from 30.5 kg instead of 0, show a suitable kink (break) on the horizontal axis.
  5. Choose a convenient scale:
    X-axis: 1 cm = 5 kg
    Y-axis: 1 cm = 2 students
  6. Draw adjacent rectangles corresponding to each class interval.
  7. Heights of the rectangles should be proportional to frequencies:
    • 30.5–35.5 → 9 students
    • 35.5–40.5 → 6 students
    • 40.5–45.5 → 15 students
    • 45.5–50.5 → 3 students
    • 50.5–55.5 → 1 student
    • 55.5–60.5 → 2 students
  8. Since it is a histogram, leave no gaps between rectangles.
🎨 SVG Diagram
Fig. 12.3
Weight Distribution of Students 0 2 4 6 8 10 12 14 16 Weight (in kg) Number of Students 9 6 15 3 1 2 30.5 35.5 40.5 45.5 50.5 55.5 60.5
👁️ Analysis of the Histogram
🔍 Finding the Modal Class

The class interval having the highest frequency is called the modal class.

Here, the highest frequency is:

\[ 15 \]

corresponding to the class interval:

\[ 40.5\text{–}45.5 \]

Therefore,

Modal Class = 40.5–45.5 kg

📋 CBSE Competency-Based Question

A student observes that the tallest rectangle in the histogram corresponds to the class interval 40.5–45.5 kg.

  1. What does the tallest rectangle represent?
  2. Can we conclude that every student weighs between 40.5 kg and 45.5 kg?
  3. What is the modal class?

Answer

  1. The interval has the highest frequency.
  2. No. It only contains the largest number of students.
  3. 40.5–45.5 kg.
⚡ Exam Tip
❌ Common Mistakes
  • Leaving gaps between histogram rectangles.
  • Drawing unequal class widths.
  • Ignoring the scale.
  • Forgetting to indicate the axis break.
  • Using category labels instead of class intervals.
  • Reading frequencies incorrectly.
📍 Key Point

Histograms are powerful tools for representing continuous grouped data. They help identify modal classes, compare frequencies, and understand distributions at a glance.

In this example, the histogram clearly shows that most students have weights between 40.5 kg and 45.5 kg, making this interval the modal class.

📊

Example 5

📋
Table of Contents
14 sections
❓ Question

A teacher wanted to analyse the performance of students in a mathematics test out of 100 marks. Since only a few students scored below 20 marks and only a few scored 70 marks or above, the marks were grouped into intervals of varying widths. Draw a histogram for the following data.

Marks Number of Students
0–20 7
20–30 10
30–40 10
40–50 20
50–60 20
60–70 15
70–100 8
💡 Concept
Concept Used
🤔 Did You Know?
Why Are Adjusted Heights Required?

In a histogram, the area of each rectangle should represent the frequency.

If one interval is twice as wide as another interval, drawing both rectangles with the same height would incorrectly increase the area of the wider interval.

Hence, when class widths differ, the rectangle heights must be adjusted so that:

\[ \text{Area of Rectangle} \propto \text{Frequency} \]

📌 Analysis of Class Widths
🔢 Formula
Formula for Adjusted Height
🗒️ Calculation Table
Marks Interval Frequency (\(f\)) Class Width Adjusted Height
0–20 7 20 \(\frac{7}{20}\times10=3.5\)
20–30 10 10 \(\frac{10}{10}\times10=10\)
30–40 10 10 \(\frac{10}{10}\times10=10\)
40–50 20 10 \(\frac{20}{10}\times10=20\)
50–60 20 10 \(\frac{20}{10}\times10=20\)
60–70 15 10 \(\frac{15}{10}\times10=15\)
70–100 8 30 \(\frac{8}{30}\times10\approx2.67\)
🔄 Step-by-Step Construction
  • 1
    Draw the X-axis and Y-axis.
  • 2
    Mark the class intervals on the horizontal axis.
  • 3
    Mark adjusted heights on the vertical axis.
  • 4
    Choose a suitable scale.
  • 5
    Draw rectangles whose widths represent class intervals.
  • 6
    Use adjusted heights instead of original frequencies.
  • 7
    Ensure adjacent rectangles touch one another.
  • 8
    Label both axes clearly.
🧩 Solution
  1. Since the class intervals are unequal, frequencies cannot be plotted directly.
  2. First calculate adjusted heights using: \[ \text{Adjusted Height} = \frac{\text{Frequency}} {\text{Class Width}} \times 10 \]
  3. Draw the histogram using these adjusted heights. The resulting graph is shown in Fig. 12.4
🎨 SVG Diagram
Fig. 12.4
Mathematics Test Performance Histogram 0 5 10 15 20 25 Adjusted Frequency (Density) Marks 0 10 20 30 40 50 60 70 80 90 100 Interval: 0-20 Frequency: 7 Adj. Frequency: 3.50 Interval: 20-30 Frequency: 10 Adj. Frequency: 10.00 Interval: 30-40 Frequency: 10 Adj. Frequency: 10.00 Interval: 40-50 Frequency: 20 Adj. Frequency: 20.00 Interval: 50-60 Frequency: 20 Adj. Frequency: 20.00 Interval: 60-70 Frequency: 15 Adj. Frequency: 15.00 Interval: 70-100 Frequency: 8 Adj. Frequency: 2.67
🔍 Analysis of the Histogram
  • The intervals 40–50 and 50–60 have the greatest frequency density.
  • Most students scored between 40 and 60 marks.
  • Very few students scored below 20 marks.
  • Very few students scored above 70 marks.
  • Student performance is concentrated around the middle score ranges.
🧠 Remember
Important Board Examination Concept
📋 CBSE Competency-Based Question

A student draws a histogram for the given data using frequencies directly as heights without calculating adjusted heights.

  1. Is the histogram correct?
  2. Why or why not?
  3. What concept has been ignored?

Answer

  1. No.
  2. Because the class intervals have different widths.
  3. Frequency density (adjusted frequency).
❌ Common Mistakes
  • Using frequencies directly when intervals are unequal.
  • Ignoring class widths.
  • Using incorrect adjusted-frequency calculations.
  • Leaving gaps between histogram rectangles.
  • Confusing bar graphs with histograms.
  • Forgetting that area represents frequency.
⚡ Exam Tip
📍 Key Point

Histograms with unequal class intervals require special treatment because rectangle areas—not merely heights—must represent frequencies.

The concept of frequency density ensures fair comparison between intervals of different widths and is one of the most important histogram concepts studied in statistics.

📊

Frequency Polygon

📋
Table of Contents
13 sections
🗺️ Overview

A frequency polygon is a graphical representation of a frequency distribution obtained by plotting points corresponding to the frequencies at the class marks (midpoints) of class intervals and joining these points with straight line segments. It provides a clear picture of the overall pattern of data and is especially useful for comparing multiple distributions on the same graph.

Frequency polygons are widely used in statistics because they occupy less space than histograms and make trends, peaks, and variations in data easier to observe. They are commonly used in education, economics, population studies, business analysis, scientific research, and social surveys.

📘 Definition
💡 Concept
🗒️ Class Mark (Midpoint)
p> The midpoint of a class interval is called its class mark.

It is calculated using:

\[ \text{Class Mark} = \frac{\text{Lower Limit}+\text{Upper Limit}}{2} \]
✏️ Example
draw a frequncy polygon chart for
Class Interval Class Mark
0–10 \(\frac{0+10}{2}=5\)
10–20 \(\frac{10+20}{2}=15\)
20–30 \(\frac{20+30}{2}=25\)
📌
Note

Why Do We Use a Frequency Polygon?

  • To study the shape of a frequency distribution.
  • To compare two or more datasets on the same graph.
  • To identify peaks and trends easily.
  • To represent grouped data without drawing large rectangles.
  • To save graphical space.
  • To obtain a smoother visual representation of data.

Illustrative Data

Class Interval Frequency
0–10 4
10–20 8
20–30 12
30–40 10
40–50 6

Calculation of Class Marks

Class Interval Frequency Class Mark
0–10 4 5
10–20 8 15
20–30 12 25
30–40 10 35
40–50 6 45
Constructing a Frequency Polygon from a Histogram
  • 1
    Draw the histogram first.
  • 2
    Locate the midpoint of the top edge of each rectangle.
  • 3
    Mark these midpoint positions.
  • 4
    Join the midpoints using straight line segments.
  • 5
    Add one imaginary class interval before the first class and one after the last class.
  • 6
    Take their frequencies as zero.
  • 7
    Join these end points to the horizontal axis.
Frequency Polygon with Histogram 0 5 10 15 20 5 15 25 Class Mark Frequency
Constructing a Frequency Polygon Without a Histogram
  • 1
    Calculate the class marks of all class intervals.
  • 2
    Draw the horizontal and vertical axes.
  • 3
    Represent class marks on the X-axis.
  • 4
    Represent frequencies on the Y-axis.
  • 5
    Plot points corresponding to each class mark and frequency.
  • 6
    Join the points using straight lines.
  • 7
    Add zero-frequency points at both ends.
Frequency Polygon 0 5 10 15 20 5 15 25 Class Mark Frequency
🗒️ Advantages of Frequency Polygon
Advantages of Frequency Polygon
  • Occupies less space than a histogram.
  • Shows the overall trend of data clearly.
  • Useful for comparing two or more distributions.
  • Easy to draw after constructing a histogram.
  • Provides a clearer picture of frequency concentration.
  • Helps identify modal regions quickly.
📊 Difference Between Histogram and Frequency Polygon
Histogram Frequency Polygon
Uses rectangles. Uses line segments.
Occupies more space. Occupies less space.
Better for showing frequencies. Better for comparing distributions.
Uses class intervals directly. Uses class marks.
Rectangles touch each other. Points are joined by straight lines.
✏️ Example
Illustrative Example

Draw a frequency polygon for the following distribution:

Class Interval Frequency
0–10 5
10–20 8
20–30 12
30–40 7
40–50 3
  1. 1
    Calculate class marks.
  2. 2
    Plot frequencies against class marks.
  3. 3
    Join the points using straight lines.
  4. 4
    Add zero-frequency end points.
Distribution Frequency Polygon 0 2 4 6 8 10 12 14 5 15 25 35 45 Class Marks (Midpoints) Frequency 5 8 12 7 3
⚡ Exam Tip
❌ Common Mistakes
  • Using class limits instead of class marks.
  • Forgetting extra zero-frequency classes.
  • Joining points with curves instead of straight lines.
  • Incorrect midpoint calculations.
  • Using unequal scales.
  • Missing axis labels.
📋 CBSE Competency-Based Question

Two schools prepared frequency polygons for marks obtained by students in a test. One graph rises sharply and then falls gradually, while the other is nearly symmetrical.

  1. Which graph indicates stronger concentration around a central value?
  2. Why are frequency polygons preferred for comparison?
  3. What information does the highest point of a frequency polygon represent?

Answer

  1. The symmetrical graph.
  2. Because multiple distributions can be drawn on the same axes.
  3. The class mark corresponding to the highest frequency.
🛠️ Application
Real-Life Applications
  • Comparing examination performance of different schools.
  • Studying population distribution.
  • Analyzing rainfall patterns.
  • Business sales analysis.
  • Monitoring manufacturing quality.
  • Scientific and medical research.
⚡ Quick Revision
  • Frequency polygons use class marks and frequencies.
  • Points are joined using straight line segments.
  • Extra zero-frequency classes are added at both ends.
  • Useful for comparing multiple distributions.
  • Occupy less space than histograms.
  • The highest point corresponds to the highest frequency.
  • Class marks are essential for accurate construction.
📊

Example 6

📋
Table of Contents
8 sections
❓ Question
Consider the marks, out of 100, obtained by 51 students of a class in a test,
Marks Number of Students
0–10 5
10–20 10
20–30 4
30–40 6
40–50 7
50–60 3
60–70 2
70–80 2
80–90 3
90–100 9
Draw a frequency polygon corresponding to this frequency distribution table.
💡 Concept
🧩 Solution
Part (i)
Calculate the Class Marks
  1. The class mark of a class interval is calculated using:
    \[ \text{Class Mark} = \frac{\text{Lower Limit}+\text{Upper Limit}}{2} \]
  2. Class Interval Frequency Class Mark
    0–10 5 5
    10–20 10 15
    20–30 4 25
    30–40 6 35
    40–50 7 45
    50–60 3 55
    60–70 2 65
    70–80 2 75
    80–90 3 85
    90–100 9 95
Part (ii)
Add Imaginary Classes
  1. To close the frequency polygon at both ends, we add one imaginary class before the first class and another after the last class.
  2. Imaginary Class Class Mark Frequency
    −10 to 0 −5 0
    100 to 110 105 0

    These additional points help the polygon meet the horizontal axis smoothly.

Part (iii)
Plot the Points
  1. Plot the following coordinate points:
  2. Point Coordinates
    A (−5, 0)
    B (5, 5)
    C (15, 10)
    D (25, 4)
    E (35, 6)
    F (45, 7)
    G (55, 3)
    H (65, 2)
    I (75, 2)
    J (85, 3)
    K (95, 9)
    L (105, 0)
Part (iv)
Join the Points
  1. Join all plotted points in sequence using straight line segments:
    A → B → C → D → E → F → G → H → I → J → K → L
  2. The resulting figure is the required frequency polygon shown in Fig. 12.7
🎨 SVG Diagram
Fig. 12.7
0 2 4 6 8 10 12 Number of Students -10 0 10 20 30 40 50 60 70 80 90 100 Marks A B C D E F G H I J K L
🔍 Interpretation
Interpretation of the Frequency Polygon
  • The highest frequency is 10 for the interval 10–20.
  • Therefore, most students scored between 10 and 20 marks.
  • Another concentration of scores is observed near 90–100 marks.
  • The least frequencies occur in the intervals 60–70 and 70–80.
  • The graph clearly shows how student performance varies across score ranges.
📌 Important Note
📋 CBSE Competency-Based Question

A student plots a frequency polygon but forgets to add the imaginary classes at both ends.

  1. Will the frequency polygon be complete?
  2. Why are imaginary classes added?
  3. What frequency is assigned to these classes?

Answer

  1. No.
  2. To ensure that the polygon begins and ends on the horizontal axis.
  3. Zero.
⚡ Exam Tip
❌ Common Mistakes
  • Using class limits instead of class marks.
  • Forgetting the imaginary classes.
  • Joining points with curves.
  • Incorrect midpoint calculations.
  • Using unequal scales.
  • Not labeling axes.
📍 Key Point

A frequency polygon is formed by plotting frequencies against class marks and joining the points with straight lines.

It provides a compact representation of grouped data and is particularly useful for comparing multiple frequency distributions on the same graph.

The addition of imaginary zero-frequency classes ensures that the polygon begins and ends on the horizontal axis, giving the graph its characteristic polygonal shape.

📊

Eample 7

📋
Table of Contents
13 sections
❓ Question
Draw a frequency polygon from the following distribution
Class Interval Frequecny \(f\)
10-20 4
20-30 8
30-40 12
40-50 10
50-60 7
60-70 4
💡 Concept
Concept Used
🔢 Formula
Formula for Class Mark
📌 Calculation of Class Marks
🤔 Did You Know?
Why Are Imaginary Classes Added?

A frequency polygon must start and end on the horizontal axis.

Therefore, we add:

  • An imaginary class before the first class: \[ 0\text{–}10 \] with frequency 0.
  • An imaginary class after the last class: \[ 70\text{–}80 \] with frequency 0.

These classes help close the frequency polygon properly.

🗺️ Roadmap
  1. Calculate class marks.
  2. Add imaginary zero-frequency classes at both ends.
  3. Draw X-axis and Y-axis.
  4. Represent class marks on the horizontal axis.
  5. Represent frequencies on the vertical axis.
  6. Plot all coordinate points.
  7. Join successive points using straight line segments.
🧩 Solution
  1. Coordinates to be Plotted
    Class Mark Frequency Coordinate
    5 0 (5, 0)
    15 4 (15, 4)
    25 8 (25, 8)
    35 12 (35, 12)
    45 10 (45, 10)
    55 7 (55, 7)
    65 4 (65, 4)
    75 0 (75, 0)
  2. Plot the points corresponding to the class marks and frequencies, then join them using straight line segments.
  3. The resulting graph is the required frequency polygon shown in Fig. 12.7-1.
🎨 SVG Diagram
Fig. 12.7-1
0 2 4 6 8 10 12 Frequency 0 10 20 30 40 50 60 70 80 Class Interval
🔍 Interpretation
Interpretation of the Frequency Polygon
  • The highest point corresponds to the interval 30–40.
  • Therefore, the interval 30–40 has the maximum frequency.
  • Most observations are concentrated around the middle intervals.
  • Frequencies decrease gradually after the modal interval.
  • The graph clearly displays the overall distribution pattern.
🗒️ Finding The Modal Class

The class interval having the highest frequency is called the modal class.

Here,

\[ \text{Maximum Frequency}=12 \]

corresponding to:

\[ 30\text{–}40 \]

Therefore,

Modal Class = 30–40

📋 CBSE Competency-Based Question

A student mistakenly plots frequencies against the lower class limits instead of class marks.

  1. Will the frequency polygon be correct?
  2. Why are class marks used instead of class limits?
  3. How would the graph change?

Answer

  1. No.
  2. Because class marks represent the centre of each interval.
  3. The entire graph would shift and become inaccurate.
⚡ Exam Tip
❌ Common Mistakes
  • Using class limits instead of class marks.
  • Ignoring imaginary classes.
  • Joining points freehand.
  • Incorrect midpoint calculations.
  • Using unequal scales.
  • Forgetting axis labels.
📍 Key Point

A frequency polygon can be drawn directly from a frequency distribution table without constructing a histogram.

The essential steps are:

  • Calculate class marks.
  • Add imaginary zero-frequency classes.
  • Plot frequencies against class marks.
  • Join the points with straight line segments.

This method is fast, accurate, and widely used in CBSE examinations and statistical analysis.

· Updated
🎓 NCERT Mathematics · Class IX

Chapter 12 · Statistics

Master data collection, organisation, graphical representation, and measures of central tendency with step-by-step guidance.

🧩 Core Concepts
Statistics is the branch of mathematics that deals with data collection, organisation, analysis, and interpretation. Chapter 12 covers the following key areas:
📊

1. Data & Its Types

Data are facts or figures collected for a specific purpose. Data can be primary (collected first-hand) or secondary (sourced from existing records).

Raw Data: Data in its original, ungrouped form, as collected.

Array: Raw data arranged in ascending or descending order.
📋

2. Frequency Distribution

A systematic way of presenting data by tallying how often each value (or group of values) occurs.

Class Interval: A range of values in a grouped distribution, e.g. 10–20.

Class Width (h) = Upper Limit − Lower Limit

Tally Marks: Used to count occurrences during data collection.
📉

3. Graphical Representations

Data is visualised using bar graphs, histograms, frequency polygons, and ogives for clearer interpretation.

Histogram: A bar graph where bars are adjacent (no gaps) — for continuous data.

Frequency Polygon: A line graph connecting mid-points of class intervals.
🔢

4. Mean (Arithmetic Average)

The most commonly used measure of central tendency. It represents the "fair share" value of a dataset.

Ungrouped: Mean = Σxᵢ / n

Grouped (Direct): Mean = Σfᵢxᵢ / Σfᵢ

Grouped (Assumed Mean): Mean = a + Σfᵢdᵢ / Σfᵢ
⚖️

5. Median

The middle value of an ordered dataset. It divides the distribution into two equal halves and is resistant to extreme values.

Odd n: Median = value at position (n+1)/2

Even n: Median = average of values at n/2 and n/2+1 positions
🏆

6. Mode

The value that appears most frequently in a dataset. A dataset can be unimodal, bimodal, or multimodal.

For ungrouped data: Simply the value with the highest frequency.

Modal Class: In grouped data, the class with the highest frequency.
🔗 Relationship Between Mean, Median & Mode

For a moderately skewed distribution, there is an empirical relationship:

Mode ≈ 3 × Median − 2 × Mean
This is an empirical (approximate) formula — valid only for moderately skewed distributions, not all datasets.
📖 Key Definitions & Terminology
The class mark is the midpoint of a class interval. It is calculated as:

Class Mark = (Lower Limit + Upper Limit) / 2

Class marks are used as representative values of each class in calculations of mean, and as x-coordinates when drawing a frequency polygon.
The cumulative frequency for a class is the running total of all frequencies up to and including that class.

It is used to draw an ogive (cumulative frequency curve) and is essential for finding the median of grouped data in higher classes.
Bar Graph: Used for discrete/categorical data. Bars are separated by gaps. Width of bars may be uniform but has no mathematical significance.

Histogram: Used for continuous grouped data. Bars are adjacent (no gaps) because class intervals are continuous. The area of each bar is proportional to the frequency.
Exclusive (continuous): Upper limit of one class = lower limit of next class. E.g., 0–10, 10–20, 20–30. The upper limit is excluded (value 10 goes into 10–20, not 0–10).

Inclusive: Both limits are included. E.g., 1–10, 11–20, 21–30. Need to convert to exclusive for histograms by adjusting boundaries.
Ungrouped data: Individual data values listed separately. Small datasets. Mean = Σx / n directly.

Grouped data: Data organised into class intervals with frequencies. Used for large datasets. Mean requires fᵢxᵢ where xᵢ is the class mark.
📐 Formula Reference Sheet
All formulas from NCERT Class IX Chapter 12. Memorise these for your examinations.

📊 Arithmetic Mean

Ungrouped Data
x̄ = (x₁ + x₂ + … + xₙ) / n
x̄ = Σxᵢ / n
Where n = number of observations
Grouped Data (Direct Method)
x̄ = Σ(fᵢ × xᵢ) / Σfᵢ
xᵢ = class mark of i-th class, fᵢ = frequency
Assumed Mean / Deviation Method
dᵢ = xᵢ − a
x̄ = a + (Σfᵢdᵢ / Σfᵢ)
a = assumed mean (usually the central class mark). This simplifies computation for large values.
Property: Effect of Adding a Constant
If y = x + k, then ȳ = x̄ + k
If y = kx, then ȳ = k × x̄
Useful shortcut for transformed datasets

⚖️ Median

Ungrouped — Odd n
Median = x at position (n+1)/2
Arrange data in ascending order first
Ungrouped — Even n
Median = ½ × [x(n/2) + x(n/2 + 1)]
Average of the two middle values

🏆 Mode & Other Quantities

Mode — Ungrouped Data
Mode = value with highest frequency
If two values share max frequency → bimodal
Class Mark (Mid-Value)
xᵢ = (Lower Limit + Upper Limit) / 2
Used as representative value for each class
Range
Range = Maximum Value − Minimum Value
Measures spread of data; sensitive to outliers
Empirical Relation
Mode ≈ 3 × Median − 2 × Mean
Valid for moderately asymmetric distributions

📏 Graphical Representation Notes

Histogram — Bar Width
Width of each bar = Class Width (h)
Height of bar ∝ Frequency (fᵢ)
For unequal class widths: height = frequency density = fᵢ/h
Frequency Polygon Points
Plot (xᵢ, fᵢ) where xᵢ = class mark
Add two extra points with f = 0 at ends
Connect the points with straight line segments
🤖 Step-by-Step Solver
Enter your data and choose an operation. The engine will solve it with full working steps — no API key required.

⚙️ Enter Your Problem

📋 Solution

🧮

Enter data on the left and click
Solve Step-by-Step

🎯 Try These Sample Problems

📝 Practice Questions
Concept-building problems with complete step-by-step solutions. All questions are original (not from the textbook).
Q1 · Mean · Easy
The ages (in years) of 10 students in a class are: 12, 13, 14, 12, 15, 13, 14, 12, 16, 15. Find the arithmetic mean of their ages.
1
Identify all values: x = {12, 13, 14, 12, 15, 13, 14, 12, 16, 15}, n = 10
2
Sum all values: Σx = 12+13+14+12+15+13+14+12+16+15 = 136
3
Apply formula: x̄ = Σx / n = 136 / 10 = 13.6
4
Verify: Mean 13.6 lies within the range [12, 16] ✓
✅ Mean age = 13.6 years
Q2 · Median · Medium
The daily wages (in ₹) of 9 labourers are: 310, 290, 275, 320, 295, 280, 310, 330, 305. Find the median wage.
1
Arrange in ascending order: 275, 280, 290, 295, 305, 310, 310, 320, 330
2
Count observations: n = 9 (odd number)
3
Find median position: (n+1)/2 = (9+1)/2 = 5th position
4
Identify 5th value: 275, 280, 290, 295, 305, 310, 310, 320, 330
✅ Median wage = ₹305
Q3 · Mode · Easy
The shoe sizes sold in a store over one week are: 7, 8, 9, 7, 8, 8, 10, 7, 8, 9, 8, 10, 8. Find the modal shoe size. What does it tell the shopkeeper?
1
Prepare frequency table:
Size 7 → 3 times | Size 8 → 6 times | Size 9 → 2 times | Size 10 → 2 times
2
Identify maximum frequency: Size 8 occurs 6 times — the highest.
3
Interpretation: Mode = 8. The shopkeeper should stock the most pairs of size 8 shoes to meet demand.
✅ Modal shoe size = 8 (occurs 6 times)
Q4 · Grouped Mean · Medium
The table below shows the number of hours students study daily. Find the mean study time using the direct method.

Hours: 0–2, 2–4, 4–6, 6–8, 8–10
Students: 4, 9, 18, 12, 7
1
Find class marks (xᵢ): 1, 3, 5, 7, 9
2
Compute fᵢxᵢ:
4×1=4 | 9×3=27 | 18×5=90 | 12×7=84 | 7×9=63
3
Sum: Σfᵢ = 4+9+18+12+7 = 50
Σfᵢxᵢ = 4+27+90+84+63 = 268
4
Apply formula: x̄ = 268 / 50 = 5.36 hours
✅ Mean study time = 5.36 hours per day
Q5 · Assumed Mean · Hard
Find the mean of the following grouped data using the Assumed Mean Method.

Marks: 20–30, 30–40, 40–50, 50–60, 60–70, 70–80
Students: 6, 10, 16, 14, 8, 6
1
Class marks xᵢ: 25, 35, 45, 55, 65, 75
2
Choose assumed mean a: Central class → a = 50 (class mark of 45–55 interval, approximately central)
3
Compute dᵢ = xᵢ − a: −25, −15, −5, +5, +15, +25
4
Compute fᵢdᵢ:
6×(−25)=−150 | 10×(−15)=−150 | 16×(−5)=−80 | 14×5=70 | 8×15=120 | 6×25=150
5
Sum: Σfᵢ = 60
Σfᵢdᵢ = −150−150−80+70+120+150 = −40
6
Apply formula: x̄ = 50 + (−40/60) = 50 − 0.67 = 49.33
✅ Mean marks = 49.33 (approximately)
Q6 · Median · Even n
The heights (in cm) of 12 plants in a garden are: 18, 26, 31, 22, 19, 27, 35, 21, 28, 24, 30, 23. Find the median height.
1
Arrange in order: 18, 19, 21, 22, 23, 24, 26, 27, 28, 30, 31, 35
2
n = 12 (even number)
3
Median = average of n/2 and n/2+1 values:
= average of 6th and 7th values
= (24 + 26) / 2 = 25 cm
✅ Median height = 25 cm
Q7 · Concept · Tricky
If the mean of 5 observations x, x+2, x+4, x+6, x+8 is 16, find the value of x. Also find all five observations.
1
Write the mean equation: (x + x+2 + x+4 + x+6 + x+8) / 5 = 16
2
Simplify numerator: 5x + 20 = 80
5x = 60 → x = 12
3
Find observations: 12, 14, 16, 18, 20
4
Verify: (12+14+16+18+20)/5 = 80/5 = 16 ✓
✅ x = 12; the five observations are 12, 14, 16, 18, 20
💡 Tips & Common Traps
What toppers know — and what most students get wrong in examinations.

✅ Smart Tips & Tricks

✓
When using the Assumed Mean method, choose a as the class mark closest to the middle of the distribution to minimise the magnitude of deviations.
✓
Always arrange data in order before finding median. Many errors happen from forgetting this step.
✓
For a frequency polygon, extend the line to zero on both ends by adding imaginary class intervals with f = 0.
✓
The mean is sensitive to extreme values; median is preferred for skewed data (e.g., income distributions).
✓
Use Σfᵢ as a quick check — it must equal the total number of observations (n).
✓
For inclusive class intervals (e.g., 1–10, 11–20), convert to exclusive form: subtract 0.5 from lower limit and add 0.5 to upper limit before drawing histograms.

❌ Common Mistakes

✗
Using class limits instead of class marks when computing mean for grouped data. Always use (L + U)/2 as xᵢ.
✗
Forgetting to sort the data before finding the median. Finding the 5th value of unsorted data gives a wrong answer.
✗
Assuming the most common digit is the mode without building a frequency table. Count carefully!
✗
Drawing a histogram with gaps between bars. Histograms must have no gaps — each bar touches the next.
✗
Confusing n/2 and (n+1)/2 for median position. Use (n+1)/2 for odd n; average of n/2 and n/2+1 for even n.
✗
Applying the empirical formula Mode = 3 Median − 2 Mean to all distributions. It only works for moderately skewed data.

🏅 Exam Strategy & Important Notes

📌
Show all working steps in exams — partial marks are awarded for correct method even if the final answer has an arithmetic error.
📌
For 3-mark grouped mean questions, always draw the fᵢ, xᵢ, fᵢxᵢ table neatly — it earns presentation marks.
📌
Units matter — always include units (kg, cm, ₹, etc.) in your final answer. Missing units lose marks.
📌
Verify reasonableness — your mean/median should lie within the range of data. If it doesn't, recheck.
🎮 Interactive Learning Modules
Choose a module to build and test your understanding interactively.
📚
ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025
Sharing this chapter
NCERT Class 9 Maths Chapter 12 Statistics Notes
NCERT Class 9 Maths Chapter 12 Statistics Notes — Complete Notes & Solutions · academia-aeternum.com
Statistics is an essential branch of mathematics that enables us to collect, organize, analyze, and interpret data meaningfully. In NCERT Class IX Chapter 12, students explore the foundational concepts of statistics, learning how to handle real-life numerical information, represent it visually with graphs, and draw insights from data. This chapter covers types of data, frequency distributions, graphical representation (such as bar graphs, histograms, and frequency polygons), mean, median, mode,…
🎓 Class 9 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
Share on
WhatsApp
Telegram
X / Twitter
Facebook
Instagram
Threads
Pinterest
LinkedIn
Quora
Discord
Reddit
Email
academia-aeternum.com/class-9/mathematics/statistics/notes/ Copy link
💡
Exam tip: Sharing chapter notes with your study group creates a reinforcement loop. Teaching a concept is the fastest path to mastering it.

Recent posts


    Fatal error: Uncaught mysqli_sql_exception: Operation not permitted in /home/u159659565/domains/academia-aeternum.com/public_html/includes/config.php:4 Stack trace: #0 /home/u159659565/domains/academia-aeternum.com/public_html/includes/config.php(4): mysqli->__construct() #1 /home/u159659565/domains/academia-aeternum.com/public_html/includes/blog_functions.php(3787): dbConnect() #2 /home/u159659565/domains/academia-aeternum.com/public_html/class-9/mathematics/statistics/notes/index.php(6129): renderCustomLinksFromDB() #3 {main} thrown in /home/u159659565/domains/academia-aeternum.com/public_html/includes/config.php on line 4