Tales by Dots and Lines — Class 8 Maths

📐 Introduction — Why Data Matters

Every day, we are surrounded by data — the runs scored by a cricket team, the temperature recorded each day, the marks obtained by students, the rainfall in different cities. But raw numbers by themselves are hard to understand. That is why we use graphs and charts to turn boring numbers into visual stories.

In this chapter, "Tales by Dots and Lines", we learn how dots (data points) and lines (connections between data points) can tell powerful stories. A single glance at a well-drawn graph can reveal trends, patterns, and comparisons that would take minutes to understand from a table of numbers.

🌎 Real-World Applications

🏏️ Weather Forecasting

Meteorologists use line graphs to show temperature changes over a week. A rising line means warming, a falling line means cooling. One glance tells you more than seven numbers.

📈 Stock Markets

Every stock price chart is a line graph with time on the x-axis and price on the y-axis. Investors make million-rupee decisions based on the shape of these lines!

🏏️ Census & Surveys

The Government of India uses bar graphs and pie charts to show population distribution, literacy rates, and economic data. The Census 2021 data is presented using these very graphs.

📊 School Reports

Your school report card, performance graphs, and attendance records all use data handling concepts from this chapter.

📚 Chapter Overview

Topic	Key Concepts
Organising Data in Tables	Tally marks, frequency tables, grouped data
Bar Graphs	Vertical and horizontal bar graphs, reading and drawing
Double Bar Graphs	Comparing two data sets side by side
Line Graphs	Showing change over time, interpreting trends
Pie Charts / Circle Graphs	Representing parts of a whole, angle calculations
Misleading Graphs	How scale, origin, and design can mislead readers
Probability Basics	Chance, outcomes, events, experimental probability

💡 Study Tip: This chapter is very visual. Practise drawing graphs neatly with a ruler and pencil. In exams, graphs are often worth 3–5 marks and require proper labelling, title, and scale.

📋 Organising Data in Tables

Before we can draw any graph, we must first organise the raw data into a clear and systematic format. The most common way to do this is by creating a frequency table.

📝 What is a Frequency Table?

A frequency table shows how often each value or category appears in a data set. The word "frequency" simply means "how many times."

Example 1: Creating a Frequency Table

Data: The favourite fruits of 20 students are: Apple, Mango, Banana, Apple, Mango, Orange, Banana, Apple, Mango, Mango, Apple, Orange, Mango, Banana, Apple, Mango, Orange, Apple, Mango, Banana.

Step 1: List all categories.
Apple, Mango, Banana, Orange

Step 2: Count using tally marks.

Fruit	Tally Marks	Frequency
Apple	┃┃┃┃ ┃	6
Mango	┃┃┃┃ ┃┃	7
Banana	┃┃┃┃	4
Orange	┃┃┃	3
Total		20

💡 Tally Mark Rule: Draw four vertical lines, then cross them with a fifth diagonal line to make a group of 5. This makes counting by fives very easy: ┃┃┃┃ = 5, then continue.

📋 Grouped Frequency Table

When data has many different values (like marks from 0 to 100), we group them into class intervals to make the table simpler.

Marks Range	Number of Students
0 – 20	3
20 – 40	8
40 – 60	12
60 – 80	10
80 – 100	7
Total	40

📚 Class Interval

Each group (e.g. 0–20, 20–40) is called a class interval. The difference between upper and lower limits is the class size (here, 20).

📋 Upper & Lower Class Limit

In the interval 20–40, 20 is the lower class limit and 40 is the upper class limit. The upper limit of one class equals the lower limit of the next.

💡 Always Check: The total of all frequencies should equal the total number of data values. If they don't match, you've made a counting error!

📊 Bar Graphs

A bar graph (or bar chart) uses rectangular bars to represent data. The length or height of each bar corresponds to the value it represents. Bars are of equal width and are separated by equal gaps.

📋 Parts of a Bar Graph

📌 Title

Every graph must have a clear title at the top that tells what the graph is about.

➡️ X-axis (Horizontal)

Usually shows the categories (e.g., fruits, months, cities). Label it clearly.

⬆️ Y-axis (Vertical)

Usually shows the values/frequency (e.g., number of students, temperature). Must start from 0 with a consistent scale.

📏 Bars

Equal width, equal spacing. Height represents the value. Can be vertical or horizontal.

✏️ How to Draw a Bar Graph

Draw the x-axis and y-axis on graph paper.
Label the x-axis with the categories.
Choose a suitable scale for the y-axis (e.g., 1 cm = 10 students).
Mark the scale on the y-axis, starting from 0.
Draw bars of equal width for each category, with the height matching the value.
Leave equal gaps between bars.
Give the graph a title.

Example 2: Draw a bar graph for the favourite fruits data.

Data: Apple = 6, Mango = 7, Banana = 4, Orange = 3

Scale: y-axis: 1 unit = 1 student (since values are small). x-axis: fruits.

Draw bars: Apple bar reaches 6, Mango bar reaches 7, Banana bar reaches 4, Orange bar reaches 3.

Observation: Mango is the most popular fruit (tallest bar), while Orange is the least popular (shortest bar).

Favourite Fruits of 20 Students

Apple

6

Mango

7

Banana

4

Orange

3

Y-axis: Number of Students | X-axis: Fruits

🔄 Horizontal Bar Graphs

In a horizontal bar graph, the bars extend from left to right. The categories are on the y-axis and the values are on the x-axis. This is useful when category names are long (e.g., city names, book titles).

📍 When to Use Vertical?

When category names are short and you want to emphasise the height (value) of each bar. Most common for comparing quantities.

📍 When to Use Horizontal?

When category names are long, or when you have many categories. Population bar charts and survey results often use horizontal bars.

💡 Reading a Bar Graph: To read the value of a bar, trace a horizontal line from the top of the bar to the y-axis (or vertical line to x-axis for horizontal bars). The point where it meets the scale axis gives you the value.

📈 Double Bar Graphs

A double bar graph uses two bars for each category to compare two related data sets side by side. Each set of bars is shown in a different colour, and a legend explains which colour represents which data set.

📋 When to Use a Double Bar Graph?

Comparing marks of a student in two different exams
Comparing sales of two products across months
Comparing boys and girls in each class
Comparing rainfall in two different years

Example 3: Marks of Riya in Mid-term vs Final Exam

Data:
English: Mid-term = 65, Final = 78
Hindi: Mid-term = 72, Final = 80
Maths: Mid-term = 58, Final = 85
Science: Mid-term = 70, Final = 76
SST: Mid-term = 68, Final = 74

Observation from graph: Riya improved in every subject from mid-term to final exam. The biggest improvement was in Maths (58 to 85, a jump of 27 marks). The least improvement was in SST (68 to 74, a jump of 6 marks).

Riya's Marks: Mid-term vs Final

Mid-term Final

Eng

Hindi

Maths

Science

SST

Scale: Height proportional to marks (max 100)

✅ Key Features of Double Bar Graphs

• Two bars per category, drawn side by side
• Different colours/patterns for each data set
• A legend (key) identifying which colour represents which data set
• Same scale for both data sets for fair comparison

💡 Reading Tips

• Compare bar heights within each category to see which is greater
• Look across categories to spot overall trends
• The gap between bars in a pair shows the difference between the two values

💡 Exam Tip: When drawing a double bar graph, always include a legend (colour key). Without a legend, the examiner cannot tell which bar represents which data set, and you may lose marks!

📉 Line Graphs

A line graph shows data points connected by straight lines. It is especially useful for showing how data changes over time (a trend). Each point on the graph represents a value, and the line connecting them shows the overall direction of change.

📋 When to Use a Line Graph?

When data changes over time (temperature, population, stock prices)
When you want to show a trend (increasing, decreasing, or constant)
When the data is continuous (the quantity exists at all times, not just at specific points)

📉 Interpreting Line Graphs

↗️ Rising Line

A line going upward from left to right indicates the value is increasing. Example: temperature rising from morning to afternoon.

↘️ Falling Line

A line going downward from left to right indicates the value is decreasing. Example: price of a product dropping during a sale.

➡️ Flat Line

A horizontal (flat) line indicates the value is constant — there is no change. Example: temperature remaining steady overnight.

⚡ Steep vs Gentle

A steep line means rapid change. A gentle (nearly flat) line means slow change. The steeper the slope, the faster the rate of change.

Example 4: Temperature of a City Over a Week

Data:
Monday: 28°C, Tuesday: 30°C, Wednesday: 33°C, Thursday: 35°C, Friday: 32°C, Saturday: 29°C, Sunday: 27°C

Reading the graph:
• Temperature increased from Monday to Thursday (line goes up).
• The highest temperature was on Thursday (35°C) — this is the peak.
• Temperature decreased from Thursday to Sunday (line goes down).
• The steepest rise was Tuesday to Wednesday (3°C jump).
• The steepest fall was Friday to Saturday (3°C drop).

Temperature Over a Week (°C)

Example 5: Company Profit Over 5 Years (in Lakhs)

Data: 2020: 15, 2021: 22, 2022: 18, 2023: 30, 2024: 35

Observations:
• Overall upward trend from 2020 to 2024 (profit growing).
• There was a dip in 2022 (from 22 to 18) — the line goes down.
• Steepest rise was 2022 to 2023 (18 to 30, a jump of 12 lakhs).
• Despite the dip, the company recovered and grew further in 2023–2024.

💡 Key Difference: Bar graphs compare different categories (fruits, cities). Line graphs show change over time (temperature over a week, sales over months). If the x-axis represents time, a line graph is almost always the better choice.

💡 Important: A line graph only makes sense when the data between the plotted points is meaningful. For example, temperature at 2:30 PM exists even if we only measured at 2 PM and 3 PM. But "favourite colour" data is not continuous — there is no "between" red and blue. So a line graph would not be appropriate for that kind of data.

🍰 Pie Charts (Circle Graphs)

A pie chart (also called a circle graph) is a circle divided into sectors (slices), where each sector represents a part of the whole. The size of each sector is proportional to the fraction or percentage it represents.

📋 Key Concepts

🟠 The Whole Circle

The entire pie represents the total (100% or 360°). Every data category gets a sector whose angle is proportional to its value.

🔹 Angle Calculation

Central angle of a category = (Value of category / Total) × 360°
Alternatively: (Percentage / 100) × 360°

Central Angle = (Value / Total) × 360° The fundamental formula for drawing a pie chart

Example 6: Monthly Spending of a Family (Total = ₹36,000)

Data:
Food: ₹10,800 | Rent: ₹9,000 | Education: ₹7,200 | Transport: ₹3,600 | Savings: ₹5,400

Step 1: Calculate each percentage.
Food = (10800/36000) × 100 = 30%
Rent = (9000/36000) × 100 = 25%
Education = (7200/36000) × 100 = 20%
Transport = (3600/36000) × 100 = 10%
Savings = (5400/36000) × 100 = 15%

Step 2: Calculate the central angle for each sector.
Food = (30/100) × 360° = 108°
Rent = (25/100) × 360° = 90°
Education = (20/100) × 360° = 72°
Transport = (10/100) × 360° = 36°
Savings = (15/100) × 360° = 54°

Verification: 108° + 90° + 72° + 36° + 54° = 360° ✅

Monthly Spending Breakdown

Food (30%)

Rent (25%)

Education (20%)

Transport (10%)

Savings (15%)

✏️ Steps to Draw a Pie Chart

Calculate the central angle for each category using the formula.
Draw a circle using a compass.
Draw a radius (a line from the centre to the edge).
Using a protractor, measure and mark the first angle from the radius.
Draw the second radius to create the first sector.
Repeat for each category, measuring angles from the previous boundary.
Label each sector with its category name and percentage.
Colour or shade each sector differently.
Add a title and legend.

📋 Reading a Pie Chart

Example 7: Reading a Pie Chart

Given: A pie chart shows the mode of transport used by 1200 students. The sector for "bus" has a central angle of 120°. How many students use the bus?

Solution:
Number of bus students = (120° / 360°) × 1200 = (1/3) × 1200 = 400 students

💡 Verification Trick: The sum of all central angles in a pie chart must always equal 360°. If they don't add up, check your calculations. Also, all percentages must add up to 100%.

💡 Pie Chart vs Bar Graph: Use a pie chart when you want to show parts of a whole (how a total is divided). Use a bar graph when you want to compare individual values. Both can represent the same data, but they tell slightly different stories.

⚠️ Misleading Graphs — Beware!

Not all graphs tell the truth! Sometimes, graphs are drawn in ways that can mislead the reader into drawing incorrect conclusions. It is important to learn how to spot misleading graphs.

🚫 Common Ways Graphs Mislead

⚠️ 1. Broken Y-axis (Not Starting from 0)

If the y-axis does not start from 0, small differences look enormous. A bar of height 98 and one of height 102 might look like a 5x difference if the axis starts at 95!

⚠️ 2. Unequal Scale

If the scale changes partway through the axis (e.g., 1 cm = 10, then suddenly 1 cm = 50), the graph becomes distorted and comparisons become unfair.

⚠️ 3. Unequal Bar Widths

If one bar is drawn wider than others, it looks more important even if its value is not higher. Fair bar graphs have equal widths.

⚠️ 4. Cherry-Picking Data

Showing only selected data points (e.g., showing only the good months) can make a declining trend look like growth. Always check if the data is complete.

⚠️ 5. 3D Effects

Three-dimensional bar graphs or pie charts can distort perception. A pie slice in the front looks bigger than one at the back, even if they represent the same value.

⚠️ 6. Misleading Pictograms

If a pictogram doubles both width and height to show a 2x increase, the area actually becomes 4x — making the increase look much bigger than it really is.

Example 8: Spotting a Misleading Graph

Scenario: A company's profit graph shows the y-axis starting at ₹95 lakhs (instead of 0). The bars show: Year 1 = ₹98 lakhs, Year 2 = ₹102 lakhs.

Misleading view: The Year 2 bar appears to be about 2.5 times the height of the Year 1 bar, suggesting massive growth.

Truth: The actual increase is only ₹4 lakhs out of ₹98 lakhs, which is roughly 4% growth — not the dramatic jump the graph suggests.

How to fix it: Start the y-axis from 0, or clearly mark the break in the axis with a zigzag symbol (~) so the reader knows the scale is not starting from 0.

💡 Critical Thinking Checklist for Graphs:
☑ Does the y-axis start from 0? If not, is the break clearly marked?
☑ Is the scale consistent (same spacing throughout)?
☑ Are the bars of equal width?
☑ Is all the relevant data shown, or only selective data?
☑ Does the title accurately describe the data?
☑ Are 3D effects distorting the perception?

🎲 Probability Basics

What is the chance of rain tomorrow? What is the likelihood of getting a 6 when you roll a die? Probability is the branch of mathematics that deals with chance and likelihood.

📚 Key Terms

🎲 Experiment

An action or process whose result is not known in advance. Example: tossing a coin, rolling a die, picking a card.

🎯 Outcome

A possible result of an experiment. When you toss a coin, the outcomes are Head (H) and Tail (T).

📋 Sample Space

The set of all possible outcomes of an experiment. For a die: {1, 2, 3, 4, 5, 6}. For a coin: {H, T}.

🌟 Event

A specific outcome or a set of outcomes we are interested in. "Getting an even number on a die" is an event = {2, 4, 6}.

📋 Experimental Probability

The experimental probability (also called empirical probability) of an event is calculated by actually performing the experiment many times and recording the results.

P(Event) = Number of times the event occurred / Total number of trials Experimental Probability Formula

Example: Tossing a Coin 50 Times

Experiment: A coin was tossed 50 times. Heads appeared 28 times and Tails appeared 22 times.

P(Head) = 28/50 = 0.56 or 56%

P(Tail) = 22/50 = 0.44 or 44%

Check: P(Head) + P(Tail) = 0.56 + 0.44 = 1.00 ✅
The sum of probabilities of all outcomes always equals 1.

🎲 Theoretical Probability

When all outcomes are equally likely, we can calculate the theoretical probability without performing the experiment:

P(Event) = Number of favourable outcomes / Total number of possible outcomes Theoretical Probability Formula (when all outcomes are equally likely)

🎲 Rolling a Die — P(6)

Favourable outcomes: {6} → 1
Total outcomes: {1,2,3,4,5,6} → 6
P(6) = 1/6 ≈ 0.167

🎲 Rolling a Die — P(even)

Favourable outcomes: {2, 4, 6} → 3
Total outcomes: 6
P(even) = 3/6 = 1/2

🃏 Drawing a Card — P(King)

Favourable outcomes: 4 kings
Total outcomes: 52 cards
P(King) = 4/52 = 1/13

🎲 Rolling a Die — P(7)

Favourable outcomes: 0 (impossible)
Total outcomes: 6
P(7) = 0/6 = 0 (impossible event)

📋 Properties of Probability

Property	Statement	Example
Range	0 ≤ P(E) ≤ 1	Probability is always between 0 and 1
Certain Event	P(certain event) = 1	P(getting a number ≤ 6 on a die) = 1
Impossible Event	P(impossible event) = 0	P(getting 7 on a standard die) = 0
Complement	P(not E) = 1 − P(E)	P(not getting 6) = 1 − 1/6 = 5/6
Sum of all outcomes	Sum of probabilities of all outcomes = 1	P(H) + P(T) = 1 for a fair coin

💡 Important: As the number of trials increases, the experimental probability gets closer to the theoretical probability. This is called the Law of Large Numbers. If you toss a coin 10 times, you might get 7 heads. But if you toss it 10,000 times, you will get very close to 5,000 heads (50%).

💡 Common Misconception: "If I got 5 heads in a row, the next toss must be tails." This is wrong! Each coin toss is independent. The coin has no memory. The probability of head on the next toss is still 1/2, regardless of what happened before. This misconception is called the Gambler's Fallacy.

✏️ NCERT-Style Worked Examples

Problem 1: The runs scored by a batsman in 6 innings are: 45, 67, 38, 72, 55, 90. (a) Draw a bar graph. (b) In which inning did he score the most? (c) What is the difference between the highest and lowest scores?

(a) Draw the x-axis with Innings 1 to 6. Y-axis with runs (scale: 1 cm = 10 runs). Draw bars of heights: 45, 67, 38, 72, 55, 90.

(b) The tallest bar is for Inning 6 with 90 runs. So the batsman scored the most in Inning 6.

(c) Highest score = 90, Lowest score = 38.
Difference = 90 − 38 = 52 runs.

Problem 2: The following table shows the number of bicycles sold by a shop over 5 months. Draw a line graph and identify the trend.

Data: Jan = 40, Feb = 55, Mar = 45, Apr = 70, May = 65

Drawing: Plot points (Jan, 40), (Feb, 55), (Mar, 45), (Apr, 70), (May, 65) and connect with straight lines.

Trend Analysis:
• Sales rose from Jan to Feb (40 to 55).
• Sales dipped in Mar (55 to 45).
• Sales peaked in Apr at 70.
• Sales slightly declined in May (70 to 65).
• Overall trend: generally upward despite fluctuations.

Problem 3: A survey of 720 students on their favourite sport gave: Cricket = 300, Football = 180, Badminton = 120, Tennis = 60, Others = 60. Draw a pie chart.

Step 1: Calculate central angles.
Cricket = (300/720) × 360° = 150°
Football = (180/720) × 360° = 90°
Badminton = (120/720) × 360° = 60°
Tennis = (60/720) × 360° = 30°
Others = (60/720) × 360° = 30°

Verification: 150° + 90° + 60° + 30° + 30° = 360° ✅

Observation: Cricket is the most popular sport (largest sector, 150°), followed by Football (90°). Together they account for 240° out of 360°, meaning two-thirds of the students prefer either Cricket or Football.

Problem 4: A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random. Find: (a) P(red) (b) P(blue) (c) P(not green) (d) P(yellow)

Total balls = 5 + 3 + 2 = 10

(a) P(red) = 5/10 = 1/2

(b) P(blue) = 3/10 = 3/10

(c) P(not green) = 1 − P(green) = 1 − 2/10 = 8/10 = 4/5

(d) P(yellow) = 0/10 = 0 (There are no yellow balls, so it is an impossible event.)

Problem 5: The pie chart of a student's daily routine shows: Sleep = 120°, School = 90°, Homework = 60°, Play = 45°, Others = 45°. If the total day is 24 hours, find the hours spent in each activity.

Formula: Hours = (Angle / 360°) × 24

Sleep = (120/360) × 24 = 8 hours
School = (90/360) × 24 = 6 hours
Homework = (60/360) × 24 = 4 hours
Play = (45/360) × 24 = 3 hours
Others = (45/360) × 24 = 3 hours

Check: 8 + 6 + 4 + 3 + 3 = 24 hours ✅

Problem 6: Two dice are thrown together. What is the probability of getting a sum of 7?

Total outcomes when two dice are thrown = 6 × 6 = 36

Favourable outcomes (sum = 7):
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

P(sum = 7) = 6/36 = 1/6

Problem 7: The marks of 30 students are given in a grouped frequency table. Draw a double bar graph to compare boys' and girls' marks.

Data:

Marks Range	Boys	Girls
0 – 20	2	1
20 – 40	4	3
40 – 60	5	6
60 – 80	3	4
80 – 100	1	1

Observations from the double bar graph:
• In the 40–60 range, girls (6) outnumber boys (5).
• In the 20–40 range, boys (4) outnumber girls (3).
• In the 80–100 range, both groups have equal numbers (1 each).
• Most students (both boys and girls) scored in the 40–60 range.

Problem 8: A spinner has 8 equal sections numbered 1 to 8. Find: (a) P(prime number) (b) P(multiple of 3) (c) P(number > 5)

Total outcomes = 8 (numbers 1 to 8)

(a) Prime numbers from 1 to 8: {2, 3, 5, 7} = 4 outcomes
P(prime) = 4/8 = 1/2

(b) Multiples of 3 from 1 to 8: {3, 6} = 2 outcomes
P(multiple of 3) = 2/8 = 1/4

(c) Numbers greater than 5: {6, 7, 8} = 3 outcomes
P(number > 5) = 3/8 = 3/8

📊 Chapter Summary

📋 Key Formulas & Concepts at a Glance

Central Angle = (Value / Total) × 360° For drawing pie charts

P(Event) = Favourable Outcomes / Total Outcomes Theoretical probability when all outcomes are equally likely

P(not E) = 1 − P(E) Complementary probability

📚 Types of Graphs — Quick Comparison

Graph Type	Best Used For	Key Feature
Bar Graph	Comparing categories	Equal-width bars, equal gaps
Double Bar Graph	Comparing two data sets	Two bars per category + legend
Line Graph	Change over time (trends)	Points connected by lines
Pie Chart	Parts of a whole	Circle divided into sectors (360°)

⚠️ Common Mistakes to Avoid

❌ Not starting y-axis from 0

Always start the y-axis from 0 in bar graphs unless you clearly mark the break.

❌ Forgetting the title & labels

Every graph needs a title, axis labels, and a scale. Missing labels = lost marks in exams!

❌ Angles not adding to 360°

In pie charts, always verify that all central angles sum to 360°. A rounding error can cause issues.

❌ Confusing P(event) range

Probability is always between 0 and 1 (inclusive). If your answer is negative or greater than 1, something is wrong.

📝 Quick Reference: Probability Facts

Fact	Value
P(certain event)	1
P(impossible event)	0
P(E) + P(not E)	1
Outcomes of a coin toss	{H, T} — 2 outcomes
Outcomes of a die roll	{1, 2, 3, 4, 5, 6} — 6 outcomes
Outcomes of two dice	6 × 6 = 36 outcomes
Cards in a standard deck	52 (4 suits × 13 cards)

💡 Exam Day Checklist:
✅ Use a ruler and pencil for all graphs.
✅ Write the title above every graph.
✅ Label both axes and mention the scale.
✅ Verify: bar widths are equal, gaps are equal.
✅ Verify: pie chart angles sum to 360°.
✅ Verify: 0 ≤ probability ≤ 1.
✅ Show all working — marks are given for steps.

🧠 Multiple Choice Questions (10 MCQs)

Click on an option to see if your answer is correct. The correct option will turn green.

Q1. The sum of all central angles in a pie chart is:
- a) 180°
- b) 360°
- c) 90°
- d) 270°
✅ Answer: (b) 360° — A pie chart is a full circle, and a full circle has 360 degrees.
Q2. Which type of graph is best for showing change over time?
- a) Bar graph
- b) Pie chart
- c) Line graph
- d) Pictogram
✅ Answer: (c) Line graph — Line graphs connect data points over time, making trends visible at a glance.
Q3. A die is rolled. What is the probability of getting an even number?
- a) 1/6
- b) 1/3
- c) 1/2
- d) 2/3
✅ Answer: (c) 1/2 — Even numbers on a die: {2, 4, 6} = 3 outcomes out of 6 total. P = 3/6 = 1/2.
Q4. In a bar graph, the bars must have:
- a) Unequal widths
- b) No gaps between them
- c) Equal widths and equal gaps
- d) Different colours only
✅ Answer: (c) Equal widths and equal gaps — This ensures a fair visual comparison between categories.
Q5. If 25% of a pie chart represents "Food," what is the central angle for "Food"?
- a) 90°
- b) 25°
- c) 180°
- d) 45°
✅ Answer: (a) 90° — Central angle = (25/100) × 360° = 90°.
Q6. The probability of an impossible event is:
- a) 0
- b) 1
- c) 0.5
- d) −1
✅ Answer: (a) 0 — An event that cannot happen has zero probability. Example: rolling a 7 on a standard die.
Q7. A double bar graph is used to:
- a) Show parts of a whole
- b) Show change over time
- c) Compare two related data sets
- d) Show the probability of events
✅ Answer: (c) Compare two related data sets — Double bar graphs show two bars per category for side-by-side comparison.
Q8. A bag has 4 red, 3 blue, and 3 white balls. What is the probability of drawing a blue ball?
- a) 4/10
- b) 3/10
- c) 3/7
- d) 1/3
✅ Answer: (b) 3/10 — Total balls = 4 + 3 + 3 = 10. P(blue) = 3/10.
Q9. A graph that can mislead the reader by not starting the y-axis from 0 is called a:
- a) Pie chart
- b) Misleading graph
- c) Line graph
- d) Double bar graph
✅ Answer: (b) Misleading graph — A broken y-axis (not starting from 0) can exaggerate differences and mislead readers.
Q10. P(E) + P(not E) equals:
- a) 0
- b) 0.5
- c) 1
- d) 2
✅ Answer: (c) 1 — The probability of an event and its complement always add up to 1.

📖 NCERT Questions & Answers

✍️ Short Answer Questions

Q1. What is a frequency table? Why is it useful?

A frequency table organises raw data by listing each category (or value) along with the number of times it occurs (its frequency). It is useful because it converts a messy list of data into a clear, organised format that makes it easy to compare values, spot the most/least common items, and prepare data for drawing graphs.
Q2. What is the difference between a bar graph and a line graph?

A bar graph uses rectangular bars to compare different categories (like favourite fruits, cities, or subjects). A line graph uses points connected by lines to show how data changes over time (like temperature, sales, or population). Bar graphs compare values at a point; line graphs show trends over a continuous period.
Q3. Define: (a) Outcome (b) Sample Space (c) Event

(a) Outcome: A possible result of a random experiment. E.g., getting "Head" when tossing a coin.
(b) Sample Space: The set of all possible outcomes. For a die: {1, 2, 3, 4, 5, 6}.
(c) Event: A specific outcome or set of outcomes of interest. E.g., "getting an even number" = {2, 4, 6}.
Q4. How do you calculate the central angle of a sector in a pie chart?

Central angle = (Value of the category / Total of all values) × 360°. For example, if Food costs ₹9000 out of a total budget of ₹36000, the central angle for Food = (9000/36000) × 360° = 90°.
Q5. A coin is tossed 100 times. It lands on heads 58 times. What is the experimental probability of getting tails?

Number of tails = 100 − 58 = 42.
P(Tails) = 42/100 = 0.42 or 42%.
Q6. Name three ways in which a graph can be misleading.

(1) Not starting the y-axis from 0 — makes small differences look large.
(2) Unequal bar widths — wider bars appear more important than they are.
(3) Cherry-picking data — showing only selected data points to present a biased picture.
Q7. What is a double bar graph? Give an example of when you would use one.

A double bar graph uses two bars for each category to compare two related data sets. For example, comparing the mid-term and final exam marks of a student in different subjects. Each subject gets a pair of bars (one for mid-term, one for final), with different colours and a legend.

📖 Long Answer Questions

Q1. The following data shows the production of wheat (in thousand tonnes) in India over 5 years: 2019 = 100, 2020 = 108, 2021 = 110, 2022 = 106, 2023 = 115. Draw a line graph and answer: (i) In which year was production highest? (ii) Was there a year when production decreased? (iii) Describe the overall trend.

Drawing the line graph: Plot points (2019, 100), (2020, 108), (2021, 110), (2022, 106), (2023, 115) on graph paper. X-axis = Years, Y-axis = Production in thousand tonnes (scale: 1 cm = 5 thousand tonnes, starting from 95 with a break mark, or from 0).

(i) Production was highest in 2023 at 115 thousand tonnes (highest point on the line).

(ii) Yes, production decreased from 2021 to 2022 (110 to 106, a drop of 4 thousand tonnes). This is shown by the line going downward.

(iii) Overall trend: The overall trend is upward (increasing production). Despite a dip in 2022, production recovered and reached a new high in 2023. The steepest rise was from 2022 to 2023 (9 thousand tonnes increase).

Q2. In a class of 40 students, the data about their favourite subjects is: Maths = 12, Science = 8, English = 6, Social Studies = 10, Hindi = 4. (a) Calculate the central angle for each subject. (b) Draw a pie chart. (c) Which subject is most popular? (d) What fraction of students prefer Maths?

(a) Central angles:
Maths = (12/40) × 360° = 108°
Science = (8/40) × 360° = 72°
English = (6/40) × 360° = 54°
Social Studies = (10/40) × 360° = 90°
Hindi = (4/40) × 360° = 36°
Verification: 108 + 72 + 54 + 90 + 36 = 360° ✅

(b) Draw a circle. Use a protractor to mark sectors of 108°, 72°, 54°, 90°, and 36° consecutively. Label and colour each sector. Add title: "Favourite Subjects of Class 8 Students."

(c) Maths is the most popular subject (largest sector, 108°, 12 students out of 40).

(d) Fraction of students who prefer Maths = 12/40 = 3/10.

Q3. A box contains 15 balls numbered 1 to 15. A ball is drawn at random. Find the probability that the number on the ball is: (a) a prime number (b) a multiple of 4 (c) an odd number (d) greater than 10 (e) divisible by both 2 and 3.

Total outcomes = 15 (balls numbered 1 to 15)

(a) Prime numbers from 1 to 15: {2, 3, 5, 7, 11, 13} = 6 numbers
P(prime) = 6/15 = 2/5

(b) Multiples of 4 from 1 to 15: {4, 8, 12} = 3 numbers
P(multiple of 4) = 3/15 = 1/5

(c) Odd numbers from 1 to 15: {1, 3, 5, 7, 9, 11, 13, 15} = 8 numbers
P(odd) = 8/15 = 8/15

(d) Numbers greater than 10: {11, 12, 13, 14, 15} = 5 numbers
P(> 10) = 5/15 = 1/3

(e) Divisible by both 2 and 3 (i.e., divisible by 6): {6, 12} = 2 numbers
P(divisible by 6) = 2/15 = 2/15

Q4. Explain with examples how graphs can be misleading. Why is it important to carefully examine the scale and axes of any graph before drawing conclusions?

Graphs can mislead in several ways:

1. Broken Y-axis: If a bar graph shows Company A's profit as ₹98 lakh and Company B's as ₹102 lakh, but the y-axis starts at 95 instead of 0, Company B's bar will look about 2–3 times taller than Company A's bar. In reality, the difference is only 4%, but the graph makes it look enormous.

2. Unequal scale: If one part of the y-axis uses 1 cm = 10 units and another part uses 1 cm = 50 units, the graph distorts the visual comparison. A sharp rise might look gentle, or a small change might look dramatic.

3. Pictogram distortion: If a pictogram uses a picture of a car to represent car sales, and doubles both the width and height of the car to show a 2x increase, the area of the car becomes 4x — making the increase look twice as large as it actually is.

4. Cherry-picking: A company might show only the months when sales were high and hide the months when sales were low, creating a false impression of constant growth.

Why it matters: Graphs are powerful communication tools used in newspapers, advertisements, reports, and social media. If we blindly trust a graph without examining its scale, origin, labels, and completeness, we can be manipulated into false beliefs. A critical reader always checks: Does the y-axis start from 0? Is the scale uniform? Is all the data shown? Are the bars equally wide? This kind of data literacy is essential in the modern world.

Q5. Compare bar graphs, line graphs, and pie charts. For each, give one situation where it would be the most appropriate choice and one situation where it would be inappropriate.

Bar Graph:
• Appropriate: Comparing the number of students in five different clubs (Drama, Music, Dance, Sports, Art). Each club is a separate category — perfect for bars.
• Inappropriate: Showing temperature changes every hour over 24 hours. The data is continuous over time, so a line graph is better.

Line Graph:
• Appropriate: Showing a patient's body temperature recorded every 2 hours for a day. Temperature changes continuously, and the line shows the trend clearly.
• Inappropriate: Showing favourite ice cream flavours of a class. "Flavour" is not a continuous or time-based variable — a bar graph or pie chart would be better.

Pie Chart:
• Appropriate: Showing how a student's 24-hour day is divided among sleep, study, play, meals, and other activities. The whole circle (24 hours = 100%) is divided into parts.
• Inappropriate: Comparing the marks of 10 students. A pie chart is about "parts of a whole," but here we want to compare individual values — a bar graph works better.

🌟 Fun Facts & Did You Know?

📊 First Bar Graph in History

The first bar graph was created by Scottish engineer William Playfair in 1786 in his book The Commercial and Political Atlas. Before Playfair, data was only shown in boring tables!

🍰 Why "Pie" Chart?

The pie chart gets its name because it looks like a pie cut into slices! Playfair also invented the pie chart in 1801. The biggest "slice" usually catches your eye first — which is exactly the point.

🎲 Dice Have Been Around for 5000 Years!

Archaeological evidence shows that dice were used in ancient Mesopotamia around 3000 BCE. The study of probability itself began much later, in the 1600s, when mathematicians Blaise Pascal and Pierre de Fermat exchanged famous letters about gambling problems.

🇮🇳 India & Data Science

India is one of the world's largest producers of data scientists! Data handling — the very skill you are learning in this chapter — is the foundation of modern fields like data science, artificial intelligence, and machine learning.

📈 Florence Nightingale & Graphs

The famous nurse Florence Nightingale used innovative pie charts (called "rose diagrams") to show that more soldiers were dying from diseases than from battle wounds. Her graphs convinced the British government to improve hospital conditions!

📉 IPL Uses Data Graphs

Every IPL match broadcast shows live line graphs of run rates, bar graphs of batsman comparisons, and pie charts of run distribution (boundaries, singles, extras). The analysis desk relies entirely on data handling concepts!

💡 Think About It: The next time you see a graph in a newspaper, an ad, or social media, ask yourself: Is this graph telling the truth? Does the y-axis start from 0? Is the data complete? You now have the tools to be a critical data reader — a superpower in today's information-rich world!

🃏 Quick Revision Flashcards

Use these flashcards for last-minute revision before your exam. Read the question, try to answer mentally, then check!

Q: What is a frequency table?

A: A table showing how many times each value/category appears in a data set.

Q: Central angle formula for pie charts?

A: (Value / Total) × 360°

Q: Bar graph vs Line graph?

A: Bar = compare categories. Line = show trend over time.

Q: P(impossible event)?

A: 0. P(certain event) = 1.

Q: P(E) + P(not E)?

A: Always equals 1.

Q: Sum of all pie chart angles?

A: 360° (a complete circle).

Q: What makes a graph misleading?

A: Y-axis not starting from 0, unequal scale, unequal bar widths, cherry-picked data.

Q: P(even) on a die?

A: {2,4,6} = 3 out of 6 = 1/2.

Q: When to use a double bar graph?

A: To compare two related data sets side by side.

Q: Who invented the bar graph?

A: William Playfair in 1786.

Q: Probability range?

A: Always between 0 and 1 (inclusive).

Q: Outcomes of two dice thrown together?

A: 6 × 6 = 36 total outcomes.

💡 Final Revision Mantra:
• Tables → Tally marks, frequency, total must match.
• Bar graphs → Equal width, equal gaps, start from 0.
• Double bar → Two bars + legend. Compare side by side.
• Line graphs → Dots + lines. Rising = increasing. Falling = decreasing.
• Pie charts → Angle = (Value/Total) × 360°. Must sum to 360°.
• Misleading → Check axis, scale, widths, data completeness.
• Probability → 0 ≤ P ≤ 1. P(E) + P(not E) = 1.

You have mastered the art of reading data stories! 💪

📊 Tales by Dots and Lines

🏏️ Weather Forecasting

📈 Stock Markets

🏏️ Census & Surveys

📊 School Reports

Example 1: Creating a Frequency Table

📚 Class Interval

📋 Upper & Lower Class Limit

📌 Title

➡️ X-axis (Horizontal)

⬆️ Y-axis (Vertical)

📏 Bars

Example 2: Draw a bar graph for the favourite fruits data.

📍 When to Use Vertical?

📍 When to Use Horizontal?

Example 3: Marks of Riya in Mid-term vs Final Exam

✅ Key Features of Double Bar Graphs

💡 Reading Tips

↗️ Rising Line

↘️ Falling Line

➡️ Flat Line

⚡ Steep vs Gentle

Example 4: Temperature of a City Over a Week

Example 5: Company Profit Over 5 Years (in Lakhs)

🟠 The Whole Circle

🔹 Angle Calculation

Example 6: Monthly Spending of a Family (Total = ₹36,000)

Example 7: Reading a Pie Chart

⚠️ 1. Broken Y-axis (Not Starting from 0)

⚠️ 2. Unequal Scale

⚠️ 3. Unequal Bar Widths

⚠️ 4. Cherry-Picking Data

⚠️ 5. 3D Effects

⚠️ 6. Misleading Pictograms

Example 8: Spotting a Misleading Graph

🎲 Experiment

🎯 Outcome

📋 Sample Space

🌟 Event

Example: Tossing a Coin 50 Times

🎲 Rolling a Die — P(6)

🎲 Rolling a Die — P(even)

🃏 Drawing a Card — P(King)

🎲 Rolling a Die — P(7)

Problem 1: The runs scored by a batsman in 6 innings are: 45, 67, 38, 72, 55, 90. (a) Draw a bar graph. (b) In which inning did he score the most? (c) What is the difference between the highest and lowest scores?

Problem 2: The following table shows the number of bicycles sold by a shop over 5 months. Draw a line graph and identify the trend.

Problem 3: A survey of 720 students on their favourite sport gave: Cricket = 300, Football = 180, Badminton = 120, Tennis = 60, Others = 60. Draw a pie chart.

Problem 4: A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random. Find: (a) P(red) (b) P(blue) (c) P(not green) (d) P(yellow)

Problem 5: The pie chart of a student's daily routine shows: Sleep = 120°, School = 90°, Homework = 60°, Play = 45°, Others = 45°. If the total day is 24 hours, find the hours spent in each activity.

Problem 6: Two dice are thrown together. What is the probability of getting a sum of 7?

Problem 7: The marks of 30 students are given in a grouped frequency table. Draw a double bar graph to compare boys' and girls' marks.

Problem 8: A spinner has 8 equal sections numbered 1 to 8. Find: (a) P(prime number) (b) P(multiple of 3) (c) P(number > 5)

❌ Not starting y-axis from 0

❌ Forgetting the title & labels

❌ Angles not adding to 360°

❌ Confusing P(event) range

📊 First Bar Graph in History

🍰 Why "Pie" Chart?

🎲 Dice Have Been Around for 5000 Years!

🇮🇳 India & Data Science

📈 Florence Nightingale & Graphs

📉 IPL Uses Data Graphs

Q: What is a frequency table?

Q: Central angle formula for pie charts?

Q: Bar graph vs Line graph?

Q: P(impossible event)?

Q: P(E) + P(not E)?

Q: Sum of all pie chart angles?

Q: What makes a graph misleading?

Q: P(even) on a die?

Q: When to use a double bar graph?

Q: Who invented the bar graph?

Q: Probability range?

Q: Outcomes of two dice thrown together?

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