Class 6 Maths – Data Handling and Presentation
Mathematics Theory • Examples • Questions • Solutions
Learn how raw information is collected, systematically organised with tally marks and frequency tables, represented visually using pictographs and bar graphs, and interpreted to discover real-world insights while ensuring visual representations remain honest and accurate.
What is Data Handling and Presentation in Class 6 Maths?
Data handling and presentation is the systematic mathematical process of collecting, organising, representing, and interpreting data so that we can clearly understand information, answer questions, compare quantities, and make well-reasoned inferences without being misled by flawed graphics.
What You Will Learn
Collecting & Organising Data
What is data, when data collection is needed, recording surveys, using tally marks, frequency tables, and ascending order.
Pictographs
Picture symbols, choosing scales (1, 5, 10, 100), interpreting partial symbols, and understanding scale limitations.
Bar Graphs
Bars of uniform width, equal spacing, why the scale must begin at zero, and comparing values and traffic patterns.
Drawing a Bar Graph
Step-by-step 10-step graph construction, choosing numerical units, budgeting, cricket run rates, and error auditing.
Artistic & Aesthetic Considerations
Horizontal vs vertical column graphs, infographics, 7 tallest mountain peaks, and detecting misleading area distortions.
Worked Examples & Puzzles
9 detailed problem walkthroughs, 10 graph practice challenges, 7 real-world investigations, and Jasprit Bumrah wicket analysis.
4.1 Collecting and Organising Data
What is data?
Data is any collection of facts, numbers, measurements, observations, or descriptions of things that convey meaningful information about those things.
The Classroom Investigation: Finding the Most Popular Game
Navya and Naresh wanted to know: "What is the most popular game in our class?" Navya asked all 30 students individually and wrote down a raw unorganised list:
Looking at this long, messy list, the children asked: "We cannot immediately see the most popular game. How do we find it?" The answer is data organisation! By grouping identical games, counting their totals, we discover: Hockey = 8, Cricket = 6, Satoliya = 5, Kabaddi = 6 (or 5), Football = 4, Badminton = 2. Hockey is the most popular game!
When Do We Need to Collect Data?
Not all questions require field data collection. We must distinguish between questions requiring empirical investigations and questions answered from existing knowledge:
- What is the most popular TV show among classmates? (Needs a survey)
- How much water is getting wasted in the locality? (Needs measurements)
- Which tree species is most common along your school route? (Needs counts)
- When did India get independence? (Historical fact: 15 August 1947)
- What is the capital of India? (Geographical fact: New Delhi)
- How many sides does a hexagon have? (Mathematical definition: 6)
Using Tally Marks & Frequency Tables
When counting observations one by one, keeping running tallies in bundles of five prevents miscounts. For every unit, write a vertical stroke (|). On the 5th item, draw a diagonal slash across the four strokes (卌).
Shri Nilesh's Sweets Example (New Year Celebration)
| Sweets Item | Tally Marks | Frequency (Students) |
|---|---|---|
| Jalebi | 卌 | | 6 |
| Gulab Jamun | 卌 |||| | 9 |
| Gujiya | 卌 卌 ||| | 13 |
| Barfi | ||| | 3 |
| Rasgulla | 卌 || | 7 |
| Total Sweets to Order | — | 38 sweets |
No! The frequency table tells the shopkeeper how many sweets to prepare (aggregate count), but it does not record which specific student chose which sweet. For distribution, a categorized name-list is needed.
Ordering Data (Ascending Order Analysis)
Sushri Sandhya asked her 27 students about their shoe sizes and recorded raw numbers on the board:
Sorting into ascending order groups identical values together:
Interactive Tally & Frequency Builder
| Category | Tally Marks Representation | Count | Adjust |
|---|---|---|---|
| Jalebi | 卌| | 6 | |
| Gulab Jamun | 卌|||| | 9 | |
| Gujiya | 卌卌||| | 13 | |
| Barfi | ||| | 3 | |
| Rasgulla | 卌|| | 7 | |
| Total Responses | — | 38 |
4.2 Pictographs
What is a pictograph?
A pictograph represents data through pictures or symbols of objects. It enables anyone to understand frequencies and compare categories with a quick glance without having to read numerical figures directly.
The Central Concept: The Scale (or Key)
A pictograph MUST always state its Key / Scale. One symbol may represent:
What Does Half a Symbol Mean?
When using a multi-unit scale, a fractional portion of a symbol represents a proportional fraction of that unit value:
- Always slept ≥ 9 hours: 5 full symbols = 5 × 10 = 50 children.
- Sometimes slept ≥ 9 hours: 2 full symbols + 1 half symbol = (2 × 10) + 5 = 25 children.
- Never slept ≥ 9 hours: 4 full symbols = 4 × 10 = 40 children.
Scale: 1 Kite Symbol (🪁) = 100 Kites.
- Rani bought 300 kites = 3 symbols.
- Poonam Ben bought 700 kites = 7 symbols.
- Poonam Ben bought more than double Rani's kites (700 > 2 × 300 = 600).
Data: A=18, B=36, C=12, D=48, E=18, F=24.
- All numbers are multiples of 6 → Best Key: 1 Dog Symbol = 6 Dogs.
- Village B = 6 symbols; Village D = 8 symbols.
- Villages B+D (36 + 48 = 84) exceed other 4 villages (18 + 12 + 18 + 24 = 72).
Interactive Pictograph Studio & Scale Selector
4.3 Bar Graphs
What is a bar graph?
A bar graph is a visual representation of data using rectangular bars of uniform width drawn with equal spacing between them. The length or height of each bar is directly proportional to the frequency or quantity of the category it represents.
| Feature | Pictograph | Bar Graph |
|---|---|---|
| Visual Medium | Pictures, icons, or symbols | Solid rectangular bars / columns |
| Scale Requirement | Key defining value per symbol | Numerical axis starting strictly at 0 |
| Handling Large Data | Can become tedious with many symbols | Effortlessly scales to thousands & crores |
| Partial Values | Awkward when values don't divide scale | Continuous height easily represents exact amounts |
In a bar graph, the entire physical length of the bar communicates quantity to our eyes. If the baseline starts at a non-zero number (e.g., 50 instead of 0), a value of 60 looks twice as tall as 55, creating a fake 100% difference when the real difference is only 9%. Starting at zero guarantees proportional honesty!
Case Study: Delhi Road Crossing Hourly Traffic (6 AM to 12 Noon)
Inferences: Peak traffic occurs between 7–8 AM (1200 vehicles) as schools and offices open. Minimum traffic is 6–7 AM (150 vehicles). Total traffic across the 6-hour observation period = 4,450 vehicles.
4.4 Drawing a Bar Graph (Step-by-Step Construction)
Drawing an accurate, high-scoring bar graph follows ten logical steps:
Worked Construction Example: Imran's Family Budget (Scale: 1 unit = ₹200)
• Electricity vs Education: ₹400 is exactly half of ₹800 (400 = 800 / 2).
• Education vs Food: Education (₹800) is less than one-fourth of food (₹800 < ₹3400 / 4 = ₹850).
Interactive Bar Graph Studio
4.5 Artistic and Aesthetic Considerations
Visual presentations should be attractive, clear, and easy to understand. However, the NCERT curriculum emphasizes a vital mathematical rule: visual aesthetics must never compromise numerical accuracy!
Vertical Bars (Column Graphs)
Ideal for quantities measured vertically upwards from the ground, such as mountain heights, tree heights, building heights, or student stature. Bars grow straight up like pillars!
Horizontal Bar Graphs
Ideal for quantities measured parallel to the ground, such as river lengths, flight distances between cities, train routes, or duration timelines.
The Seven Continent Summits (Tallest Mountains on Earth)
| Continent | Peak Name | Height (Meters) |
|---|---|---|
| Asia | Mount Everest | 8,848 m |
| South America | Aconcagua | 6,962 m |
| North America | Denali | 6,194 m |
| Africa | Kilimanjaro | 5,895 m |
| Europe | Mount Elbrus | 5,642 m |
| Antarctica | Vinson Massif | 4,892 m |
| Australia | Mount Kosciuszko | 2,228 m |
When designers replace simple rectangular bars with triangular mountain illustrations or 3D cones, a serious mathematical deception occurs:
The 2D Area Distortion Flaw: When a triangle is made taller, its base also widens. The human brain perceives the area of the shape (Area = 0.5 × base × height). Thus, Everest (8848m) looks four to six times larger than Elbrus (5642m), even though Elbrus is actually 64% of Everest's height (5642 × 2 = 11284 > 8848).
Aesthetic designs that distort proportionality communicate false information! Always prefer equal-width bars for honest mathematical comparison.
Accurate vs Misleading Visualizer
Data Handling and Presentation – Worked Examples
Thirty students stated their favourite sweet: 6 Jalebi, 9 Gulab Jamun, 13 Gujiya, 3 Barfi, 7 Rasgulla. Construct a frequency table with tally marks.
• Jalebi (6): 5 + 1 = 卌 |
• Gulab Jamun (9): 5 + 4 = 卌 ||||
• Gujiya (13): 5 + 5 + 3 = 卌 卌 |||
• Barfi (3): 3 = |||
• Rasgulla (7): 5 + 2 = 卌 ||
• Total check: 6 + 9 + 13 + 3 + 7 = 38 sweets.
A class of 27 students has shoe sizes: 3, 3, 3, 4(x9), 5(x10), 6(x4), 7(x1). How many students wear sizes larger than 4?
• Students wearing size 5 = 10
• Students wearing size 6 = 4
• Students wearing size 7 = 1
• Sum = 10 + 4 + 1 = 15 students.
In a pictograph where 1 Triangle (▲) = 10 Children, the 'Sometimes' sleep category has 2 complete triangles and 1 half-triangle. How many children does this represent?
• 2 complete triangles = 2 × 10 = 20 children.
• 1 half-triangle = 10 ÷ 2 = 5 children.
• Total = 20 + 5 = 25 children.
The Mudhol Hound dog counts in 6 villages are 18, 36, 12, 48, 18, and 24. What is the most practical scale to avoid messy fractional symbols?
• 18 = 6 × 3
• 36 = 6 × 6
• 12 = 6 × 2
• 48 = 6 × 8
• 24 = 6 × 4
• All values are exact whole-number multiples of 6.
A bar graph shows hourly traffic in Delhi: 6-7 AM (150), 7-8 AM (1200), 8-9 AM (1000). What is the total traffic from 8 AM to 10 AM if 9-10 AM is 800?
• 8-9 AM bar height = 1000 vehicles
• 9-10 AM bar height = 800 vehicles
• Total = 1000 + 800 = 1800 vehicles.
Smriti's scores in 8 matches are: 80, 50, 10, 100, 90, 0, 90, 50. Why is 1 unit = 1 run unsuitable, and what scale should be chosen?
• A scale of 1 unit = 1 run requires 100 physical divisions, which is unnecessarily huge.
• A scale of 1 unit = 10 runs requires only 10 grid steps (0, 10, 20... 100).
• All scores (0, 10, 50, 80, 90, 100) fall perfectly on exact multiples of 10.
If Imran's family spends ₹3400 on food and the scale is 1 unit = ₹200, what is the required bar height in grid units?
• Expenditure on Food = ₹3400
• Scale = ₹200 per unit length
• Bar Height = 3400 ÷ 200 = 17 grid units.
Rani bought 300 kites and Poonam Ben bought 700 kites. Rukhsana claims Poonam Ben bought more than double Rani's kites. Is she correct?
• Rani's kites = 300
• Double Rani's purchase = 2 × 300 = 600 kites
• Poonam Ben's purchase = 700 kites
• Since 700 > 600, Poonam Ben exceeded double.
An infographic illustrates Mount Everest (8848m) and Mount Elbrus (5642m) using triangles. Everest appears twice as tall as Elbrus. Why is this mathematically false?
• Mount Elbrus height = 5642 m
• Twice Elbrus height = 5642 × 2 = 11,284 m
• Mount Everest height is only 8848 m, which is far less than 11,284 m.
Roll a Die 30 Times Experiment
In NCERT Chapter 4, students roll a die 30 times and record every outcome. Use this live simulator to observe how raw outcomes transform into tally marks and frequencies:
Frequency Reasoning: Total Wickets Calculation
Faiz prepared a frequency table of wickets taken by Jasprit Bumrah in his last 30 matches. Mayank says: "To find total wickets, just add 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28." Is Mayank right?
| Wickets Taken (x) | Matches (Frequency f) | Total Wickets in these matches (x × f) |
|---|---|---|
| 0 wickets | 2 matches | 0 × 2 = 0 |
| 1 wicket | 4 matches | 1 × 4 = 4 |
| 2 wickets | 6 matches | 2 × 6 = 12 |
| 3 wickets | 8 matches | 3 × 8 = 24 |
| 4 wickets | 3 matches | 4 × 3 = 12 |
| 5 wickets | 5 matches | 5 × 5 = 25 |
| 6 wickets | 1 match | 6 × 1 = 6 |
| 7 wickets | 1 match | 7 × 1 = 7 |
| Total | 30 matches | 90 total wickets! |
Graph Practice: 10 Core Challenges
Middle School Ginnori recorded books borrowed: Mon(5), Tue(4), Wed(2), Thu(0), Fri(5), Sat(8). On which day were the minimum books borrowed, and what was the weekly total?
Villages have dog counts: A=18, B=36, C=12, D=48, E=18, F=24. How many symbols represent Village D if 1 symbol = 6 dogs?
Magan Bhai sold 100, 250, 300, 450, and 700 kites. Why is 1 symbol = 100 kites chosen instead of 1 symbol = 5 kites?
In Lakhanpal's graph for Classes I to VIII (3, 5, 4, 2, 0, 1, 5, 7), which class had 100% full attendance?
On Pooja's railway board, Vidisha (24 tickets) is 6 units high and Jabalpur (20 tickets) is 5 units high. What is the scale?
If Sagar had 16 tickets sold, how high should its bar be drawn using the scale of 1 unit = 4 tickets?
Samantha counted: Caterpillars 10, Mites 6, Beetles 5, Butterflies 3, Grasshoppers 2. Which scale best fits a small notebook page?
Project Tiger tracked tigers: 2006 (1400), 2010 (1700), 2014 (2200), 2018 (3000), 2022 (3700). Why would drawing 2014 shorter than 2010 be an error?
In Imran's budget, Food is ₹3400 and Education is ₹800. Is Education less than one-fourth of Food?
Why does a 3D cylindrical bar graph or widening triangle mislead viewers when representing 1D numerical data?
Try It Yourself: 7 Real-World Data Surveys
Determine the top sports preference among 30 peers.
1. Ask each student • 2. Record with tally marks • 3. Compile frequency table • 4. Construct vertical column graph.
Discover colour distribution in your section.
1. List Red, Blue, Green, Yellow, Purple • 2. Count tallies • 3. Calculate percentage of class for each colour.
Count Neem, Peepal, Banyan, and Gulmohar trees.
1. Tally trees during commute • 2. Find most/least common species • 3. Formulate environmental conclusions.
Audit shoe sizes to find mode and threshold totals.
1. Record sizes • 2. Sort into ascending order • 3. Calculate how many wear sizes > 4.
Observe Bikes, Cars, Buses, Cycles, Auto-rickshaws.
1. Stand safely by road • 2. Make tally bundles • 3. Compare two-wheelers vs four-wheelers.
Count occurrences of letters 'c', 'e', 'i', 'r', 'x'.
1. Cut a 100-word news snippet • 2. Tally each letter • 3. Verify that 'e' is most frequent and 'x' least.
Track saplings planted during first week of July.
1. Record daily plantation logs • 2. Draw bar graph • 3. Analyze weekend vs weekday patterns.
Challenge Questions: Deep Conceptual Inquiries
On a bar graph representing daily absenteeism across classes, Class V has a bar height of zero. Does this mean data is missing, or what does it mathematically convey?
Why does choosing a scale of 1 symbol = 10 students fail when a class has 33 or 27 students?
If two students scored 95 and 90 marks, and the vertical scale starts at 85 instead of 0, how does the visual comparison distort reality?
Why is total wickets obtained from a frequency table by summing (wickets × matches) rather than simply adding the numbers in the wickets column?
When mountain peaks are represented as triangles rather than uniform-width columns, why does the graphic exaggerate the height difference?
Why is it conventional to use vertical bars for mountain heights and horizontal bars for river lengths?
If a student redraws a bar graph changing the scale from 1 unit = 5 to 1 unit = 10, does the underlying data or relative ratio between bars change?
Why does 'What is the capital of India?' NOT require data collection, while 'What is the most popular TV show among your friends?' DOES require data collection?
Where Do We Use Data Handling in Real Life?
Studying hourly traffic volumes at busy road crossings to optimize signal timers.
Plotting run rates, bowling figures, and wicket frequencies across 30 matches.
Tracking Project Tiger population growth from 1400 (2006) to 3700 (2022).
Comparing monthly household spending on rent, food, education, and transport.
Key Takeaways
Frequently Asked Questions about Data Handling & Presentation
Computational Thinking with Data Handling and Presentation
Data handling develops foundational skills essential to Computational Thinking, such as tabular structuring, frequency binning, algorithmic sorting, visual information encoding, and detecting statistical bias or distortion in datasets. Explore the Class 6 Computational Thinking & AI lessons.