Data Handling and Presentation
Class 6 Maths (Ganita Prakash) — Chapter 4
If you ask your classmates their favourite colours, you get a list of colours. If you weigh every student in your class, you get a set of weight measurements. Both are examples of data.
What is Data?
Any collection of facts, numbers, measures, observations or descriptions that gives us information about something is called data.
Any collection of facts, numbers, measures, observations or descriptions that gives us information about something is called data.
We live in an age of information, where data is shown to us in many interesting ways. In this chapter we learn how to display data, read it correctly, and draw useful conclusions from it.
4.1 Collecting and Organising Data
Suppose you want to find the most popular game in a class. A plain list of names and games is hard to read directly. So we organise the data into a table using tally marks.
How tally marks work: Draw one line | for each item. When the count reaches 5, draw a diagonal line across the previous four to make a bundle: ||||⁄ (a group of 5). This makes counting fast.
A bundle of five tally marks
Frequency
The number of times a value or category appears in the data is called its frequency. For example, if 8 students like hockey, the frequency of hockey is 8.
The number of times a value or category appears in the data is called its frequency. For example, if 8 students like hockey, the frequency of hockey is 8.
Example Frequency Table (Sweets)
A teacher, Shri Nilesh, asked students their favourite sweet and recorded it with tally marks.
| Sweet | Tally Marks | No. of Students (Frequency) |
|---|---|---|
| Jalebi | ||||⁄ | | 6 |
| Gulab jamun | ||||⁄ |||| | 9 |
| Gujiya | ||||⁄ ||||⁄ ||| | 13 |
| Barfi | ||| | 3 |
| Rasgulla | ||||⁄ || | 7 |
Arranging in order: Data can also be sorted in ascending order (smallest to largest). This makes it easy to spot the smallest value, largest value and repeated values quickly.
Figure it Out — Page 75
Q1. What would you do to find the most popular game among the classmates?
Ans. Arrange the collected names and games into a table using tally marks. Then count the frequency of each game and pick the one with the highest frequency. (Other ways, like grouping students by game, also work.)
Q2. What is the most popular game in their class?
Ans. Hockey, with a frequency of 8.
Q4. Tick (✔) where data collection is needed and cross (✗) where it is not.
Ans.
- a. Most popular TV show among classmates — ✔ (must ask people)
- b. When did India get independence — ✗ (already a known fact)
- c. How much water is wasted in her locality — ✔ (must measure/observe)
- d. Capital of India — ✗ (already a known fact)
Figure it Out — Page 76 (Sweets Table)
Q1. Complete the table.
Ans.
- a. Jalebi = 6
- b. Barfi = 3
- c. Gujiya = 13
- d. Rasgulla = 7
- e. Gulab jamun = 9
Q2. Is this table enough to give each sweet to the correct student?
Ans. No. The table only shows how many students chose each sweet, not which student chose which sweet. A better way is to group students name-wise under their chosen sweet.
Figure it Out — Page 77 (Shoe Sizes)
Shoe sizes in ascending order: 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7
Q1.
Ans.
- a. Largest shoe size = 7
- b. Smallest shoe size = 3
- c. Students wearing size 5 = 10
- d. Students wearing sizes larger than 4 = 15 (ten 5s + four 6s + one 7)
Q2. How did arranging in ascending order help?
Ans. In order, equal values sit together, so the frequency of each size is easy to count and the smallest/largest values are seen at once.
Q3. Are there other ways to arrange the data?
Ans. Yes. The data can be arranged in a frequency table.
4.2 Pictographs
Pictograph
A pictograph shows data using pictures or symbols of objects. It lets us understand data at a quick glance, without writing many numbers.
A pictograph shows data using pictures or symbols of objects. It lets us understand data at a quick glance, without writing many numbers.
Imp for exams
Every pictograph needs a scale (key) that tells what one symbol represents — for example, one symbol = 1 student, or one symbol = 10 children. A half symbol represents half of that scale value.
Every pictograph needs a scale (key) that tells what one symbol represents — for example, one symbol = 1 student, or one symbol = 10 children. A half symbol represents half of that scale value.
Example: Sleep Survey (1 symbol = 10 children)
| Response | Symbols (▲ = 10 children) | Number |
|---|---|---|
| Always | ▲▲▲▲▲ | 5 × 10 = 50 |
| Sometimes | ▲▲▾ | 20 + 5 = 25 |
| Never | ▲▲▲▲ | 4 × 10 = 40 |
So 50 children always slept at least 9 hours, 25 sometimes did, and 40 never did (meaning 40 children always slept less than 9 hours).
Imp for exams
Pictographs get tricky when data is large or when a number is not an exact multiple of the scale. Example: if a symbol = 10 students, showing 33 or 27 students is hard because you cannot neatly draw parts like 3 or 7.
Pictographs get tricky when data is large or when a number is not an exact multiple of the scale. Example: if a symbol = 10 students, showing 33 or 27 students is hard because you cannot neatly draw parts like 3 or 7.
Figure it Out — Page 83
Q. Problems in making a pictograph if the class has 33 or 27 students?
Ans. With one symbol = 10 and a half symbol = 5, we can only show multiples of 5. Numbers like 3 or 7 cannot be drawn exactly, so the pictograph cannot show 33 or 27 accurately.
Figure it Out — Page 84 (Books Borrowed)
Symbols per day (1 symbol = 1 book): Monday 6, Tuesday 4, Wednesday 2, Thursday 0, Friday 5, Saturday 7.
Q1.
Ans.
- a. Minimum books borrowed: Thursday (0 books)
- b. Total books during the week = 6 + 4 + 2 + 0 + 5 + 7 = 24
- c. Maximum books: Saturday. Possible reason: Sunday is a holiday, so students borrow more to read at home.
Figure it Out — Page 85 (Kites, 1 symbol = 100 kites)
| Shopkeeper | Kites Sold | Symbols (◆ = 100) |
|---|---|---|
| Chaman | 250 | 2½ (◆◆◇) |
| Rani | 300 | 3 (◆◆◆) |
| Rukhsana | 100 | 1 (◆) |
| Jasmeet | 450 | 4½ (◆◆◆◆◇) |
| Jetha Lal | 250 | 2½ (◆◆◇) |
| Poonam Ben | 700 | 7 (◆◆◆◆◆◆◆) |
Q2.
Ans.
- a. Rani = 300 kites = 3 symbols
- b. Maximum kites: Poonam Ben (700)
- c. Jasmeet (450) sold more than Chaman (250), so Jasmeet
- d. Poonam Ben = 700, double of Rani (300) = 600. Since 700 > 600, Yes, she is correct. (700 = 2 × 300 + 100)
4.3 Bar Graphs
Bar Graph
A bar graph shows data using rectangular bars of equal width. The length or height of each bar shows the frequency of that category. Bars have equal gaps between them.
A bar graph shows data using rectangular bars of equal width. The length or height of each bar shows the frequency of that category. Bars have equal gaps between them.
When data is large, a pictograph becomes slow and messy. A bar graph is faster and clearer. Here is the bar graph for the number of students absent in each class (1 unit = 1 student): Class I = 3, II = 5, III = 4, IV = 2, V = 0, VI = 1, VII = 5, VIII = 7.
No. of students absent in each class (1 unit = 1 student)
Figure it Out — Page 86
Answer using the bar graph.
Ans.
- Q1. In Class 2, 5 students were absent.
- Q2. Maximum absent: Class 8 (7 students).
- Q3. Full attendance (0 absent): Class 5.
Reading a horizontal bar graph (traffic example): Vehicles at a Delhi crossing (1 unit = 100 vehicles): 6–7 a.m. ≈ 150, 7–8 a.m. = 1200 (maximum), 8–9 a.m. = 1000, 9–10 a.m. = 800, 10–11 a.m. = 700, 11–12 noon ≈ 600.
Figure it Out — Page 88 (Traffic)
Q1. Total cars from 6 a.m. to noon?
Ans. 150 + 1200 + 1000 + 800 + 700 + 600 = 4450 cars.
Q2–Q4 (Reasoning).
Ans.
- Q2. Traffic is low from 6–7 a.m. because most people are still at home; work and school have not started.
- Q3. Traffic is heaviest 7–8 a.m. as people rush to offices, schools and shops at the same time.
- Q4. After 8 a.m. traffic falls each hour because most people have already reached their destinations.
Population example: India's population (in crores) grew 36 → 44 → 54 → 68 → 84 → 102 across the decades 1951 to 2001. Using a scale of 1 unit = 10 crore makes the bars easy to draw.
4.4 Drawing a Bar Graph
Steps to Draw
Draw two lines — one horizontal, one vertical
→
Write categories on the horizontal line, equally spaced
→
Choose a suitable scale on the vertical line
→
Draw equal-width bars with heights matching the frequency
Imp for exams
The markings on the axis must always start from zero. All bars must have the same width and equal gaps between them. Choose the scale using the smallest and largest values so the graph fits the page neatly.
The markings on the axis must always start from zero. All bars must have the same width and equal gaps between them. Choose the scale using the smallest and largest values so the graph fits the page neatly.
Choosing a Scale (Runs Example)
Smriti's runs: 80, 50, 10, 100, 90, 0, 90, 50. Since values go up to 100, using 1 unit = 1 run is tiring. Instead use 1 unit = 10 runs.
Expenditure Example — Calculating Bar Heights (1 unit = ₹200)
| Item | Expenditure (₹) | Height |
|---|---|---|
| House rent | 3000 | 3000 ÷ 200 = 15 units |
| Food | 3400 | 3400 ÷ 200 = 17 units |
| Education | 800 | 800 ÷ 200 = 4 units |
| Electricity | 400 | 400 ÷ 200 = 2 units |
| Transport | 600 | 600 ÷ 200 = 3 units |
| Miscellaneous | 1200 | 1200 ÷ 200 = 6 units |
Figure it Out — Page 93 (Expenditure)
Answer using the expenditure bar graph.
Ans.
- Q1. Most = Food (₹3400); second most = House rent (₹3000).
- Q2. Electricity = ₹400, Education = ₹800. Since 400 is half of 800 — Yes.
- Q3. Education = ₹800, Food = ₹3400. One-fourth of 3400 = 850. Since 800 < 850 — Yes, education is less than one-fourth of food.
Figure it Out — Page 93 Q1 (Insects Bar Graph)
Data: Mites 6, Caterpillars 10, Beetles 5, Butterflies 3, Grasshoppers 2.
Bar graph of insects Samantha saw (1 unit = 1 insect)
Figure it Out — Page 94 Q2 (Tickets Sold)
Data: Vidisha 24, Jabalpur 20, Seoni 16, Indore 28, Sagar 16.
Q2.
Ans.
- a. Vidisha = 24 tickets.
- b. Jabalpur = 20 tickets.
- c. Vidisha bar = 6 units for 24 tickets, so scale = 24 ÷ 6 = 1 unit = 4 tickets.
- d. Sagar = 16 tickets = 4 units tall.
- e. Vertical axis marks: 0, 4, 8, 12, 16, 20, 24, 28.
- f. Seoni bar is correct; Indore bar is incorrect and should be redrawn at 28 (7 units).
Figure it Out — Page 94 Q3 (Transport)
Q3.
Ans.
b. Used the most: Bike.
c. To collect this data: make a two-column table (transport, tally); watch the road and mark a tally for each vehicle that passes; then count the tallies to get frequencies.
| Means of Transport | Frequency |
|---|---|
| Bike | 13 |
| Car | 6 |
| Bicycle | 8 |
| Auto Rickshaw | 8 |
| Scooter | 9 |
| Bus | 4 |
| Bullock Cart | 2 |
c. To collect this data: make a two-column table (transport, tally); watch the road and mark a tally for each vehicle that passes; then count the tallies to get frequencies.
Figure it Out — Page 95 Q4 (Rolling a Die)
Q4.
Ans. This is a hands-on task. Roll a die 30 times, mark each result (1–6) with tally marks, then count. From your own table: (a) find the number with the smallest count, (b) the number with the largest count, and (c) any numbers that appear an equal number of times. Results vary for each student.
Figure it Out — Page 95 Q5 (Bumrah's Wickets)
| Wickets Taken | No. of Matches | Total Wickets |
|---|---|---|
| 0 | 2 | 0 |
| 1 | 4 | 4 |
| 2 | 6 | 12 |
| 3 | 8 | 24 |
| 4 | 3 | 12 |
| 5 | 5 | 25 |
| 6 | 1 | 6 |
| 7 | 1 | 7 |
| Total | 30 | 90 |
Q5.
Ans.
- a. It shows how many matches Bumrah took each number of wickets in.
- b. A good title: "Wickets Taken by Jasprit Bumrah in his Last 30 Matches".
- c. Interesting point: he took 7 wickets in only 1 match (rare and high).
- d. He took 4 wickets in 3 matches.
- e. No. Adding 0+1+2+…+7 only adds the wicket values once. It ignores how many matches each value happened in.
- f. Multiply each "wickets" by its "number of matches", then add them all: total = 90 wickets.
Figure it Out — Page 96 Q6 (Tractors)
1 symbol = 1 tractor. Village A = 6, B = 5, C = 8, D = 3, E = 6.
Q6.
Ans.
- a. Smallest number of tractors: Village D (3).
- b. Most tractors: Village C (8).
- c. Village C – Village B = 8 – 5 = 3 more tractors.
- d. Village D = 3, Village E = 6, and 3 is half of 6 — Yes, she is right.
Figure it Out — Page 97 Q7 (Girl Students)
1 symbol = 4 girls. Class 1 = 24, 2 = 18, 3 = 20, 4 = 14, 5 = 10, 6 = 16, 7 = 12, 8 = 6.
Q7.
Ans.
- a. Least girls: Class 8 (6 girls).
- b. Class 6 (16) – Class 5 (10) = 6.
- c. Class 2 has 18 (four full + one half). Adding 2 more makes 20, so the half symbol becomes a full symbol.
- d. Class 7 = 3 symbols × 4 = 12 girls.
Figure it Out — Page 98 Q8 (Mudhol Hounds)
Village A = 18, B = 36, C = 12, D = 48, E = 18, F = 24.
Q8.
Ans.
- a. A good scale is 1 symbol = 6 dogs, because every value is a multiple of 6.
- b. Village B = 36 ÷ 6 = 6 symbols.
- c. B + D = 36 + 48 = 84. Other four (A+C+E+F) = 18+12+18+24 = 72. Since 84 > 72, Kamini is right.
Figure it Out — Page 98 Q9 (Free-time Activity)
Scale 1 unit = 5 students. Playing 45, Reading 30, Watching TV 20, Music 10, Painting 15.
Preferred free-time activity (1 unit = 5 students)
Q9.
Ans. Other than playing, the most preferred activity is Reading story books (30 students).
Figure it Out — Page 99 Q10 (Saplings)
Mon 52, Tue 40, Wed 30, Thu 40, Fri 50, Sat 60, Sun 40.
Q10.
Ans.
- a. Wednesday + Thursday = 30 + 40 = 70.
- b. Whole week = 52+40+30+40+50+60+40 = 312.
- c. Greatest: Saturday (60); least: Wednesday (30). Reasons may include more helpers on weekends, weather, or how many students were present. You can check by asking or recording the reason each day.
Figure it Out — Page 99 Q11 (Tigers in India)
| Year | Number of Tigers (approx.) |
|---|---|
| 2006 | 1400 |
| 2010 | 1700 |
| 2014 | 2200 |
| 2018 | 3000 |
| 2022 | 3700 |
Q11. Find and fix the mistakes in the graph.
Ans. The bars for 2006, 2010, 2014 and 2018 are drawn at wrong lengths. They should be corrected so their lengths match 1400, 1700, 2200 and 3000. (The 2022 bar for 3700 is correct.)
4.5 Artistic and Aesthetic Considerations
Besides being correct, a graph should look neat, fit the space, and be easy to read. We can control this by choosing the right scale, using colours, and picking the right bar direction.
Column Graph
A bar graph with vertical bars is also called a column graph (like the pillars that hold up a roof). Heights (like mountains) look natural with vertical bars.
A bar graph with vertical bars is also called a column graph (like the pillars that hold up a roof). Heights (like mountains) look natural with vertical bars.
Choosing bar direction:
- Heights (e.g. mountains, tallest person) → use vertical bars, since height goes upward.
- Lengths along the ground (e.g. river lengths, distances) → use horizontal bars.
Tallest Mountains (Example)
| Continent | Mountain | Height (m) |
|---|---|---|
| Asia | Everest | 8848 |
| South America | Aconcagua | 6962 |
| North America | Denali | 6194 |
| Africa | Kilimanjaro | 5895 |
| Europe | Elbrus | 5642 |
| Antarctica | Vinson Massif | 4892 |
| Australia | Koscuiszko | 2228 |
Column graph comparing the tallest peaks of each continent
Infographics
Infographic
When a graph is decorated with pictures and colours to make it more attractive, it is called an infographic. It communicates information quickly in a pleasing way.
When a graph is decorated with pictures and colours to make it more attractive, it is called an infographic. It communicates information quickly in a pleasing way.
Imp for exams
Fancy pictures can mislead. If mountain triangles are drawn wider when they are taller, a viewer may think taller mountains are also wider — extra information that is not true. Also, if bars do not start at zero, differences may look bigger or smaller than they really are. Always be careful so a graph does not fool the reader.
Fancy pictures can mislead. If mountain triangles are drawn wider when they are taller, a viewer may think taller mountains are also wider — extra information that is not true. Also, if bars do not start at zero, differences may look bigger or smaller than they really are. Always be careful so a graph does not fool the reader.
Example of misleading scale: Everest (8848 m) may be drawn to look "twice as tall" as Elbrus (5642 m), but 5642 × 2 = 11284, which is far more than 8848. So Everest is not twice as tall as Elbrus — the picture would be wrong.
Figure it Out — Page 103
Q1. Heights of the tallest persons in each class — vertical or horizontal bars?
Ans. Use vertical bars, because height is measured upward from the ground, so vertical bars feel natural for comparing heights.
Q2. Longest rivers on each continent — vertical or horizontal bars?
Ans. Use horizontal bars, because a river's length runs along the ground, so horizontal bars represent length more naturally. (Both types can be used, but horizontal fits better here.)