NCERT Solutions Class 6 Maths Chapter 4 Data Handling and Presentation

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 6 · M AT H S

NCERT Solutions

Chapter 4: Data Handling and
Presentation

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

74 – 106 14 46 English

Solutions, notes, sample papers & more at 65 pages

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 4: Data Handling and
Presentation
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 4 Data Handling and Presentation from
the NCERT textbook Ganita Prakash. Every Figure it Out question from pages 75 to 103 is solved, along with
all the in-text questions, using tally marks, frequency tables, pictographs and clearly labelled bar graphs.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 74 – 106

SECTIONS QUESTIONS

14 46

MEDIUM

English

Figure it Out — Pages 75 & 76
Section 4.1 Collecting and Organising Data

Q1 What would you do to find the most popular game among Naresh’s and Navya’s
classmates?

ANSWER

Turn the long list into a frequency table. Read the list once, put a tally mark against a game
every time a child chooses it, and then count the tally marks.

1. Write the different games in one column — kabaddi, hockey, cricket, satoliya (pittu), football,
badminton.
2. Go down the list name by name. For every child put one mark | against the game they
named.
3. After four marks draw the fifth across them — |||| — so the marks always come in bundles
of five and are easy to count.
4. Count the marks of each game. That count is the frequency of the game.
5. The game with the biggest frequency is the most popular game.

Page 1 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

GAME TALLY MARKS NUMBER OF STUDENTS (FREQUENCY)

Kabaddi |||| | 6

Hockey |||| ||| 8

Cricket |||| | 6

Satoliya (Pittu) |||| 5

Football |||| 4

Badminton || 2

Total — 31

6 + 8 + 6 + 5 + 4 + 2 = 31 students — the whole class is counted, so nobody has been left out

or counted twice.

Why the table helps: the raw list has 31 names jumbled together, so the answer
cannot be seen in it. The table throws away the names (which we do not need here)
and keeps only the game and its count — and every child lands in exactly one row.

Q2 What is the most popular game in their class?

ANSWER

Hockey is the most popular game — 8 of the 31 students chose it.

Hockey 8 > Kabaddi 6 = Cricket 6 > Satoliya 5 > Football 4 > Badminton 2

Page 2 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Favourite game of the class
9
8
8

7
Number of students

6 6
6
5
5
4
4

3
2
2

1

0
Hockey Kabaddi Cricket Satoliya Football Bad-
(Pittu) minton

Game

The same frequency table drawn as a bar graph. The tallest bar — hockey — answers the question at
a glance.

Did you notice? Navya had guessed that cricket was the most popular game, but
the data says cricket was chosen by only 6 students. A guess is not data — this is
exactly why Naresh and Navya went and asked everybody.

Q3 Try to find out the most popular game among your classmates.

ANSWER

This is an activity for your own class. Do it exactly the way Naresh and Navya did:

1. Frame one clear question — “Which one game do you like the most?” Every child answers
with exactly one game.
2. Collect — go to each classmate and note the answer, or let each child raise a hand when
their game is called out.
3. Organise — put a tally mark against the game each time it is named, in bundles of five.
4. Count — write the frequency beside each game and add them up. The total must equal the
number of children in your class.
5. Conclude — the game with the largest frequency is the most popular game of your class.

Page 3 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
m.
GAME TALLY MARKS FREQUENCY

m ase
Cricket
.co
|||| |||| || 12
a g l
se m
g l a
Kho-kho
a
|||| ||| 8

Kabaddi |||| | 6

co m
Football ||||
e m . 4 ag
g l as
Total —
a 30

c o m
In this sample class of 30 children, cricket would be the most popular game. Your own class will
.
s em
almost certainly give a different answer — that is the whole point of collecting data.
m a
. co a gl
a s em it yourself: always add the frequencies at the end. If the total is not the same
Check

agl as the number of students you asked, a name has been missed or counted twice.
m a s
em
.co agl
l a s ✓
Pari wants to respond to the questions given below. Put a tick ( ) for the questions
agout data collection and put a cross (✗) for the questions
Q4
where she needs to carry
where she doesn’t need to collect data. Discuss your answers in the classroom. a.

. c om
What is the most popular TV show among her classmates? b. When did India get

s e m d. What is the
independence? c. How much water is getting wasted in her locality?
m of India? a
.co agl
capital

a s em
agl ANSWER
se m
QUESTION TICK /
com
WHY
g l a
m . a
ase
CROSS

a. Most popular TV show ag✓l The answer is different in every class, and it changes with
among her classmates time. Only her own classmates can tell her — so she must
ask them and count.
co m
m .
as e
com
✗
l
b. When did India get It is a fixed historical fact — 15 August 1947. It is already

. a g
m
independence? recorded in books; asking people would only give

l a se opinions, some of them wrong.

ag ✓ .c
m
c. How much water is Nobody has this number ready. She must go around, look

m add up — that is data collection. a s e
agl
getting wasted in her at leaking taps and pipes, measure or count them, and
locality?
. co
s✗e m
l a
ag
d. What is the capital of A fixed fact — New Delhi. One reliable source is enough.
India?

com
m .
m as e
.co


a g l Page 4 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

The rule behind the answers: collect data when the answer is not already known, or
when it can be different for different people, places or times. If the answer is a
single settled fact, just look it up — collecting data would be a waste of effort.

Try This: Sort these into the two groups — “How many left-handed children are in
my class?”, “How many days are there in a leap year?”, “Which bus route is the most
crowded near my school?”, “What is the boiling point of water?” (First and third need
data; the other two do not.)

Figure it Out — Pages 76 & 77

Page 5 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Section 4.1 Tally Marks and Frequency

Q1 Complete the table to help Shri Nilesh to purchase the correct numbers of sweets:
a. How many students chose jalebi? b. Barfi was chosen by ____ students? c. How
many students chose gujiya? d. Rasgulla was chosen by ____ students? e. How many
students chose gulab jamun?

SWEETS TALLY MARKS NO. OF
STUDENTS

Jalebi 6

Gulab 9
jamun

Gujiya

Barfi

Rasgulla

The tally-mark table Shri Nilesh made. The last three counts are still blank.

ANSWER

Page 6 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Count each bundle of five first, then the loose marks.

Gujiya: |||| |||| ||| = 5 + 5 + 3 = 13

Rasgulla: |||| || = 5 + 2 = 7

Barfi: ||| = 3

SWEETS TALLY MARKS NO. OF STUDENTS

Jalebi |||| | 6

Gulab jamun |||| |||| 9

Gujiya |||| |||| ||| 13

Barfi ||| 3

Rasgulla |||| || 7

Total — 38

a. 6 b. 3 c. 13 d. 7 e. 9

6 + 9 + 13 + 3 + 7 = 38 — so there are 38 students in Shri Nilesh's class, and he must buy 38

sweets in all.

Page 7 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Sweet preferences of students
14 13

12
Number of students

10 9

8 7
6
6

4 3

2

0
Jalebi Gulab Gujiya Barfi Rasgulla
jamun

Sweets

The same frequency table as a bar graph. Gujiya (13) is the class favourite; barfi (3) is the least liked.

Why tally marks are written in fives: our eyes cannot count a long row of single
strokes without mistakes. Bundles of five can be counted 5, 10, 15, … at a glance —
the same reason we count notes and coins in bundles.

Page 8 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
m.
Is the above table sufficient to distribute each type of sweet to the correct student?
e
Q2

m l as
.co
Explain. If it is not sufficient, what is the alternative?

a g
se m
g l a
a SWEETS TALLY MARKS NO. OF

com
STUDENTS

e m . ag
Jalebi
as
6

agl

co m
em.
m l as9
.co
m jamun
Gulab
a g
l a se
a g
m a s
.co agl
Gujiya

se m
g l a
a
Barfi

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl
Rasgulla

co m
m .
m l a se
m .co The tally-mark table Shri Nilesh made. The lasta g counts are still blank.
se
three

g l a
a c
m .
m a s e
co agl
ANSWER

m .
No, the table is not sufficient.
as e
a g l

co m
m .
m ase
.co


a g l Page 9 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

The table tells Shri Nilesh how many students chose each sweet, but not which student chose
which sweet. It says 13 gujiyas are needed — it does not say whose 13 they are. The names
were thrown away while making the table.
The alternative: keep a second, name-wise record — a list of the students under each sweet.

SWEET NAMES OF THE STUDENTS WHO CHOSE IT COUNT

Jalebi Aarav, Meena, Sahil, … 6

Gulab jamun Fatima, Ravi, Neha, … 9

Gujiya Kabir, Sita, Amrit, … 13

Another simple way is to write the sweet next to each name in the attendance register, exactly as
Navya's first list did for games.

The big idea: a frequency table is a summary. Summarising always throws some
detail away. So before you summarise, ask what question you want to answer —
“how many to buy?” needs only counts, but “who gets what?” needs the names too.

Figure it Out — Pages 77 – 79
Section 4.1 Arranging Data in Order

MATH TALK

Q1 Help her to figure out the following: a. The largest shoe size in the class is ______. b.
The smallest shoe size in the class is ______. c. There are ______ students who wear
shoe size 5. d. There are ______ students who wear shoe sizes larger than 4.

ANSWER

Sushri Sandhya has already written the 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

Page 10 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

SHOE SIZE TALLY MARKS NUMBER OF STUDENTS

3 ||| 3

4 |||| |||| 9

5 |||| |||| 10

6 |||| 4

7 | 1

Total — 27

a. 7 — the last number in the ordered list.
b. 3 — the first number in the ordered list.
c. 10 students wear size 5.
d. 15 students wear a size larger than 4.

Sizes larger than 4 = size 5 + size 6 + size 7

= 10 + 4 + 1 = 15

Shoe sizes in the class
12

10
10
9
Number of students

8

6

4
4
3

2
1

0
3 4 5 6 7

Shoe size

The shoe-size data as a bar graph. Size 5 is the most common; only one student wears size 7.

Page 11 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Check it yourself: 3 + 9 + 10 + 4 + 1 = 27, and the board has 3 rows of 9 numbers =
27 readings. The counting is correct.

Q2 How did arranging the data in ascending order help to answer these questions?

ANSWER

Ordering does three things at once:

The smallest and the largest jump out. In an ordered list they are simply the first and the
last number — 3 and 7. In the jumbled board you would have to scan all 27 numbers, twice.
Equal values come together. All the 5's stand side by side, so counting them is one quick
sweep instead of hunting all over the board.
“More than” questions become easy. Everything larger than 4 lies to the right of the last 4
— you just count the tail of the list.

Why it works: ordering does not change the data at all — the same 27 readings are
there. It only changes the arrangement, and a good arrangement makes the answer
visible instead of hidden.

Math Talk: Ask your friend to find the largest number in a jumbled list of 27
numbers, and time it. Then give the same list in order. Discuss why the second is so
much faster.

Q3 Are there other ways to arrange the data?

ANSWER

Yes — many. Each way suits a different question:

Descending order — 7, 6, 6, 6, 6, 5, 5, … Useful when you want the biggest values first.
Frequency table with tally marks — the shortest form of all: five rows instead of 27
numbers (the table in question 1 above).
Pictograph — one symbol for each student (or for every 2 students) against each size.
Bar graph — a bar for each size, its height showing how many students wear it.
Grouped table — “small sizes (3–4)”, “large sizes (5–7)”, if only the broad picture is needed.

Page 12 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Which one to pick: the frequency table is best for exact counts, the bar graph for
comparing at a glance, and the ordered list for finding the smallest, the largest or
the middle value. The data never changes — only the way we look at it.

Q4 Write the names of a few trees you see around you. When you observe a tree on the
way from your home to school (or while walking from one place to another place),
record the data and fill in the following table: a. Which tree was found in the
greatest number? b. Which tree was found in the smallest number? c. Were there
any two trees found in the same numbers?

TREE NO. OF TREES

Peepal

Neem

…

….

The table given in the book, to be filled in with your own count.

ANSWER

This is a survey you do outside the classroom. Carry a small notebook, and put one tally mark
every time you pass a tree of that kind.

TREE TALLY MARKS NO. OF TREES

Peepal |||| ||| 8

Neem |||| |||| || 12

Banyan |||| 4

Mango |||| || 7

Gulmohar |||| 4

Total — 35

For this sample walk:

Page 13 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

a. The neem was found in the greatest number — 12 trees.
co m
b. The banyan and the gulmohar were found in the smallest number — 4 trees each.
e m.
m l as
.co a g
c. Yes — banyan and gulmohar have the same count, 4 each.

a s em
l
Your own table will have different trees and different numbers, and that is perfectly correct —
aganswer
the belongs to your route.

. com
Tip: fix the route and the day before you start, and count only trees on one side of
ag
m data cannot be compared with a friend's
the road. If the plan keeps changing,ethe
s
a
data.
agl

co m
se m.
o m l a
Take a blank piece of paper and paste any small news item from a newspaper. Each
.c may use a different article. Now, prepare a table aongthe piece of paper as
Q5

m
se given below. Count the number of each of the letters ‘c’, ‘e’, ‘i’, ‘r’, and ‘x’ in the words
student
a
agl of the news article, and fill in the table. a. The letter found the most number of

s
times is ______. b. The letter found the least number of times is ______. c. List the five
om a
. c agl
letters ‘c’, ‘e’, ‘i’, ‘r’, ‘x’ in ascending order of frequency. Now, compare the order of

a s em Is your order the same or nearly the same as
your list with that of your classmates.

a l likely to get the order ‘x, c, r, i, e’.) Why do you think this
theirs? (Almost everyonegis
is the case? d. Write the process you followed to complete this task. e. Discuss with
your friends the processes they followed. f. If you do this task with another news
com
m .
e
item, what process would you follow?

m l as
m .co a g
l a se
ag LETTER C E I R X ANY OTHER LETTER OF

m
YOUR CHOICE

a se
. com a g l
m
Number of times found in

ase
the news item

a gl
The table given in the book, to be filled in from your news item.

co m
m .
m as e
.co a g l
emHere is a filled table for one short news item of about 120 words:
ANSWER

a s
agl
.c
LETTER C E I R X ANY OTHER LETTER (A)
s e m
m a
e m . co agl
as
Number of times found in the news item 18 72 46 40 1 55

a g l
a. Among the five given letters, e was found the most number of times.

m
b. x was found the least number of times.

. co
e m
m l as
.co a g
Page 14 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

c. Ascending order of frequency: x, c, r, i, e (1, 18, 40, 46, 72).

Why everybody gets nearly the same order: the order does not depend on which
news item you pick. It depends on the English language itself. e is the commonest
letter in English — it sits in the, he, be, she, and in almost every ending like -ed, -es, -
ment. x is the rarest — very few words use it at all. So any piece of ordinary English,
long enough, will show the same pattern. This is a real and useful fact: it is exactly
how secret codes are cracked, and how the letters were arranged on old typewriters
and keyboards.

d. The process I followed:

1. Pasted the news item and read it once to see how long it was.
2. Drew the table with one column for each letter.
3. Read the article one word at a time, and put a tally mark in the right column for each letter
found — bundling the marks in fives.
4. Counted both capital and small forms of a letter together (E and e), and did not count the
headline separately from the text.
5. Counted the tally marks, wrote the frequencies, and arranged them in ascending order.

e. Different friends work differently — some go letter by letter (reading the whole article once
for ‘c’, once for ‘e’, and so on), some strike off each letter with a pencil as they count, and some
cut the article into lines and count line by line. Going through the article once with all five
columns open is the fastest and makes fewer mistakes.
f. For another news item I would follow the same process — that is the whole point of writing
the steps down. Only then can the two results be compared fairly. I would also note the number
of words in each article, because a longer article naturally gives bigger counts.

In-text Questions — Page 80
Section 4.2 Pictographs

Q1 Which mode of travel is used by the most number of students? Which mode of
travel is used by the least number of students?

ANSWER

First read the key printed beside the pictograph: one symbol = 1 student. So nothing has to be
multiplied here — just count the symbols in each row.

Page 15 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Modes of Travelling — Number of Students
= 1 student

Private car 4

Public bus 5

School bus 11

Cycle 3

Walking 7

The pictograph of page 79, read row by row. The school bus row is the longest and the cycle row the
shortest.

MODES OF TRAVELLING SYMBOLS COUNTED NUMBER OF STUDENTS

Private car 4 4

Public bus 5 5

School bus 11 11

Cycle 3 3

Walking 7 7

Total 30 30

School bus 11 > Walking 7 > Public bus 5 > Private car 4 > Cycle 3

The school bus is used by the most number of students (11), and the cycle by the least
number of students (3).

Total students surveyed = 4 + 5 + 11 + 3 + 7 = 30

Why a pictograph answers so fast: your eye compares lengths of rows far more
quickly than it compares numbers written in a table. The longest and the shortest
row are spotted before you count anything at all.

Page 16 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q2 What is the number of children who always slept at least 9 hours at night?

ANSWER

The key says one symbol stands for 10 children. The row ‘Always’ has 5 complete symbols.

5 symbols × 10 children each = 50 children

How often children slept at least 9 hours
= 10 children

Always 50 children

Sometimes 25 children

Never 40 children

Nand Kishor's pictograph. One circle = 10 children; a half circle = 5 children.

Answer: 50 children.

Tip: never read a pictograph without reading its key first. The very same picture
would mean 5 children if the key said “= 1 child”, and 500 if it said “= 100 children”.

Q3 How many children sometimes slept at least 9 hours at night?

ANSWER

The row ‘Sometimes’ has 2 complete symbols and 1 half symbol.

2 complete symbols = 2 × 10 = 20 children

1 half symbol = half of 10 = 5 children

Total = 20 + 5 = 25 children

Answer: 25 children.

Page 17 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Why a half symbol is allowed: the scale here is 10, so numbers like 25 are not exact
multiples of the scale. Drawing half a picture is the pictograph's way of showing “half
of 10”, that is 5. Values such as 23 or 27 could still not be shown neatly — that is a
real limitation of pictographs.

Q4 How many children always slept less than 9 hours each night? Explain how you got
your answer.

ANSWER

“Always slept less than 9 hours” is the same as “never slept at least 9 hours”. So we read the row
‘Never’.

Never → 4 complete symbols

4 × 10 = 40 children

Answer: 40 children.

How the answer was found: the trick is in the words, not in the arithmetic. A child
who never sleeps 9 hours or more must be sleeping less than 9 hours on every night
— that is, always less than 9 hours. So the two sentences describe exactly the same
group of children.

Check it yourself: 50 + 25 + 40 = 115 children answered Nand Kishor's survey in all.

In-text Questions — Pages 82 & 83
Section 4.2 Drawing a Pictograph

MATH TALK

Q1 If they want to show their data through a pictograph, where they also use one
symbol for each student, as Lakhanpal did, what are the challenges they might
face?

ANSWER

Lakhanpal's data was small — only 3, 5, 4, 2, 0, 1, 5, 7 students, that is 27 symbols altogether.

Page 18 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
Students absent in each class
e m.
m l as
.co a g
= 1 student

a s em VIII
gl
Class 7
a Class VII 5

co m
. ag
Class VI 1

e m
Class V
g l as 0

Class IV a 2

Class III 4
co m
em.
com as
Class II 5

. Class I a g l
m
ase
3

agl Lakhanpal's pictograph: students absent in each class, one symbol for one student. Class V has no
m
symbol at all — nobody was absent.
a s
m .co agl
Jarina and Sangita's data is much bigger:
l a se
a g
30 + 35 + 20 + 25 + 30 + 25 + 30 + 20 = 215 students
co m
m .
m as e
.co
So with one symbol per student they would face these challenges:
a g l
a s em
l
Far too many symbols — 215 of them have to be drawn, one by one.
ag It takes very long, and the hand gets tired; the last symbols start looking different from the
first ones.
se m
com g l a
. a
It will not fit — a row of 35 symbols runs off the edge of the notebook page.
m
ase
Counting becomes hard — to read “35” the reader must count 35 tiny pictures, and may

agl
easily miscount. The picture then answers no faster than a plain table.

The cure is a bigger scale. Jarina used one symbol for 5 students, so only 43 symbols were

co m
.
needed:

em
m l as
m .co a g
l a se
ag
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 19 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Students present — one symbol for 5 students
= 5 students

Class VIII 20

Class VII 30

Class VI 25

Class V 30

Class IV 25

Class III 20

Class II 35

Class I 30

Jarina's pictograph with the key “1 symbol = 5 students”. Class II, with 35 students, needs 7 symbols.

Sangita went further and used one symbol for 10 students, with a half symbol for 5:

Students present — one symbol for 10 students
= 10 students

Class VIII 20

Class VII 30

Class VI 25

Class V 30

Class IV 25

Class III 20

Class II 35

Class I 30

Sangita's pictograph with the key “1 symbol = 10 students”. 25 students are shown as 2 symbols and a
half.

The lesson: the scale must suit the size of the data. Small numbers → 1 symbol = 1
unit. Large numbers → 1 symbol = 5, 10, 100 … units. But the scale must also divide
the data neatly, otherwise the next problem appears — see the next question.

Page 20 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q2 What could be the problems faced in preparing such a pictograph, if the total
number of students present in a class is 33 or 27?

ANSWER

Sangita's key is “one symbol = 10 students”, and a half symbol = 5 students. Now try 33 and 27:

33 = 3 × 10 + 3 → 3 symbols and 3 more students

27 = 2 × 10 + 7 → 2 symbols and 7 more students

To show the extra 3 students we would need three-tenths of a symbol, and for 7 students
seven-tenths of a symbol. Such tiny fractions of a picture cannot be drawn — and certainly
cannot be read back correctly.
A half symbol is no help: half of 10 is 5, which is neither 3 nor 7.
Two different readers would guess two different numbers from the same part-symbol. The
pictograph stops being accurate.

What to do instead:

Choose a scale that divides all the values — with 33 and 27 present, 1 symbol = 1 student is
exact (but long); if all the numbers were multiples of 3, a scale of 3 would work.
Or give up the pictograph and draw a bar graph, where a bar can be drawn to any height, so
33 and 27 are shown exactly.

Math Talk: Which scale would you pick for 12, 18, 24 and 30? (A scale of 6 works
perfectly — 2, 3, 4 and 5 symbols.) Now try it for 12, 19, 24 and 30. Discuss what goes
wrong.

Figure it Out — Pages 83 – 85
Section 4.2 Reading and Making Pictographs

Q1 The following pictograph shows the number of books borrowed by students, in a
week, from the library of Middle School, Ginnori: a. On which day were the
minimum number of books borrowed? b. What was the total number of books
borrowed during the week? c. On which day were the maximum number of books
borrowed? What may be the possible reason?

ANSWER

The key is “one symbol = 1 book”, so the number of symbols in a row is the number of books.

Page 21 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Books borrowed from the library in a week
= 1 book

Monday 5

Tuesday 4

Wednesday 2

Thursday 0

Friday 5

Saturday 8

The library pictograph read row by row. Thursday's row is empty and Saturday's is the longest.

DAY MON TUE WED THU FRI SAT

Books borrowed 5 4 2 0 5 8

a. Thursday — its row is the shortest of all, so the fewest books were borrowed that day.
b. Add the six days:

5 + 4 + 2 + 0 + 5 + 8 = 24 books

c. Saturday — the longest row, 8 books.
Possible reason: Sunday is a holiday. Children borrow books on Saturday so that they have
something to read at home on the long free day, and there is no school library open on Sunday
to borrow from. Thursday may be low because of a class test, extra homework or a games
period that day.

Remember: a reason like this is a guess (a hypothesis), not a fact. To be sure, one
would have to collect more data — for example, the library record of several weeks.

Page 22 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q2 Magan Bhai sells kites at Jamnagar. Six shopkeepers from nearby villages come to
purchase kites from him. The number of kites he sold to these six shopkeepers are
given below — Chaman 250, Rani 300, Rukhsana 100, Jasmeet 450, Jetha Lal 250,
Poonam Ben 700. Prepare a pictograph using the symbol to represent 100 kites.
Answer the following questions: a. How many symbols represent the kites that Rani
purchased? b. Who purchased the maximum number of kites? c. Who purchased
more kites, Jasmeet or Chaman? d. Rukhsana says Poonam Ben purchased more
than double the number of kites that Rani purchased. Is she correct? Why?

ANSWER

With the key “one symbol = 100 kites”, divide every number by 100. A half symbol stands for 50
kites.

250 ÷ 100 = 2½ 300 ÷ 100 = 3 100 ÷ 100 = 1

450 ÷ 100 = 4½ 250 ÷ 100 = 2½ 700 ÷ 100 = 7

Kites sold by Magan Bhai
= 100 kites

Chaman 250

Rani 300

Rukhsana 100

Jasmeet 450

Jetha Lal 250

Poonam Ben 700

The pictograph of kites sold. One full circle = 100 kites, one half circle = 50 kites.

a. Rani bought 300 kites, and 300 ÷ 100 = 3 symbols.
b. Poonam Ben — 700 kites, the longest row.
c. Jasmeet — 450 kites against Chaman's 250, so Jasmeet bought 200 kites more.
d. Yes, Rukhsana is correct.

Page 23 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
em.
Double of Rani's kites = 2 × 300 = 600
m l as
Poonam Ben bought 700
m .co a g
l a se
g
700 > 600, and 700 − 600 = 100 kites more than double
a

. com
Check it yourself: Magan Bhai sold 250 + 300 + 100 + 450 + 250 + 700 = 2050 kites
ag
a s em
in all — that is 20½ symbols on the pictograph.

agl

co m
In-text Questions — Page 86
e m.
m l as
.co a g
Section 4.3 Bar Graphs

a s em
aglQ1 In Class 2, ___________ students were absent that day.
m a s
ANSWER
m.co agl
l a se
The scale of the graph is 1 unit length = 1 student, so the height of a bar can be read straight
off the vertical line. ag

co m
m .
m
No. of students absent in each class
as e
.co 8 a g l
a s em 7

agl 7

m
6
se
Number of students

5
com
5
g l a
4. a
5

e m
as
agl
4
3
3

m
2

. co
2

em
as
1
m l
.co
1

a g
em
0
s
0
l a
ag
I II III IV V VI VII VIII

.c
s e m
m a
Class

e m . co agl
as
Lakhanpal's bar graph. The bar for Class II reaches the line marked 5.

a g l

co m
m .
m ase
.co


a g l Page 24 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Bar for Class II reaches the gridline 5 → 5 students were absent

Answer: 5.

Tip: to read a bar, put a ruler flat on its top edge and slide your eye left to the vertical
line. The equally spaced horizontal lines are drawn exactly for this — every gap
stands for the same number.

Q2 In which class were the maximum number of students absent? ___________

ANSWER

Look for the tallest bar in the graph.

Class VIII → 7 students absent — the tallest bar of all

(next comes Class II and Class VII with 5 each)

Answer: Class VIII (8th class), with 7 students absent.

Why the eye finds it instantly: all the bars have the same width and the same
spacing, and they all start from 0. So only their heights differ, and the tallest bar can
mean nothing except the largest number.

Q3 Which class had full attendance that day? ___________

ANSWER

Full attendance means nobody was absent, that is, the number of absent students is 0. So look
for the class with no bar at all.

Class V → 0 students absent → full attendance

Answer: Class V (5th class).

Page 25 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Did you notice? A value of 0 is shown by no bar, not by a missing column. The place
for Class V is still kept on the horizontal line, with the same spacing as every other
class — otherwise the graph would tell a lie about how many classes there are.

Total absent that day = 3 + 5 + 4 + 2 + 0 + 1 + 5 + 7 = 27 students

Figure it Out — Page 88
Section 4.3 Reading a Bar Graph

Q1 How many total cars passed through the crossing between 6 a.m. and noon?

ANSWER

Read the length of each bar first. One unit of length stands for 100 vehicles.

Traffic at a Delhi crossing (6 a.m. – 12 noon)

11–12 600

10–11 700

9–10 800

8–9 1000

7–8 1200

6–7 150

0 200 400 600 800 1000 1200
Number of vehicles

The traffic bar graph. The bars are horizontal, so their lengths are compared side by side.

TIME INTERVAL 6–7 7–8 8–9 9–10 10–11 11–12

Vehicles 150 1200 1000 800 700 600

Page 26 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

150 + 1200 + 1000 + 800 + 700 + 600

= (150 + 1200) + (1000 + 800) + (700 + 600)

= 1350 + 1800 + 1300

= 4450 vehicles

Answer: about 4450 vehicles passed through the crossing in those six hours.

Tip: the word “about” matters. The 6–7 a.m. bar is a little short of 200, so we read it
as roughly 150. A bar graph gives quick, close values — not exact counts to the last
vehicle.

Q2 Why do you think so little traffic occurred during the hour of 6–7 a.m., as compared
to the other hours from 7 a.m.–noon?

ANSWER

Only about 150 vehicles passed between 6 and 7 a.m. — roughly one-eighth of the 7–8 a.m.
traffic. Sensible reasons:

At 6 a.m. most people are still at home — just waking up, getting ready, having breakfast.
Schools, offices, shops and markets have not opened yet, so there is nowhere to travel to.
School buses, office buses and delivery vans start their trips later.
The few vehicles at that hour belong to early workers — milk vans, newspaper vans,
vegetable trucks going to the mandi, and people on the night shift returning home.

The general idea: traffic follows the daily routine of people. Wherever a large
number of people have to be at the same place at the same time, the roads become
crowded just before that time.

Q3 Why do you think the traffic was the heaviest between 7–8 a.m.?

ANSWER

Between 7 and 8 a.m. about 1200 vehicles crossed — the longest bar on the graph. This is the
morning rush hour:

Schools start at about 8 a.m., so school buses, vans, autos, cycles and parents dropping
children are all on the road together.

Page 27 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Offices, factories and shops open between 9 and 10, and people in a big city like Delhi need
an hour or more to reach, so they leave home at this time.
Everyone is trying to reach before a fixed time, so all these journeys are squeezed into the
same single hour.

Did you know? A second rush hour usually appears in the evening, between about 5
and 8 p.m., when the same people travel back home. This survey stopped at noon,
so that peak does not appear in the graph.

Q4 Why do you think the traffic was lesser and lesser each hour after 8 a.m. all the way
until noon?

ANSWER

Look at the bars after the peak — they get shorter and shorter, step by step:

7–8: 1200 → 8–9: 1000 → 9–10: 800 → 10–11: 700 → 11–12: 600

Reasons for this steady fall:

By 9 or 10 o'clock most people have already reached school, office or work, so those
journeys are over and are not repeated.
Whoever is left travels for occasional reasons — shopping, hospital visits, deliveries, visiting
relatives. Such trips are far fewer, and they are spread out over the whole day.
Nobody has to reach anywhere at 11 a.m. sharp, so there is no new crowd of vehicles at any
one moment.

Fall each hour: 1200 → 1000 (−200), 1000 → 800 (−200), 800 → 700 (−100), 700 → 600

(−100)

Notice the shape: the traffic drops quickly just after the rush is over, and then
settles down slowly. It never becomes zero, because a city always has some travel
going on.

In-text Questions — Page 89

Page 28 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

Section 4.3 Choosing a Scale
co m
em.
m l as
Q1
.co
On the basis of this bar graph, what may be a few questions you may ask your
m a g
l a se
friends?

a g

com
. ag
110
102

e m
as
100

a g l
Population of India in crores

90 84
80
68
m
70

co
m.
60 54

as e
com l
50 44

. a g
m
40 36

a s e
agl
30
20

s
10

m a
.co 1971 agl
0

se m
1951 1961 1981 1991 2001

g l a Years
a

co m
.
The bar graph on page 88 — population of India, in crores, in each decade.

e m
m l as
.co a g
a s em
gl
ANSWER
a Good questions are the ones a reader can answer from the graph itself. Here are some you can

se m
com a
ask:

. a g l
m
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 29 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Population of India in crores
110 102
100
Population of India in crores

90 84
80
68
70
60 54
50 44
40 36
30
20
10
0
1951 1961 1971 1981 1991 2001

Years

Population of India, 1951 to 2001. One unit length stands for 10 crore people.

Reading questions — What was the population of India in 1971? (54 crore) In which year did
it first cross 100 crore? (2001)
Comparing questions — Was the population in 1991 more than double that of 1951? (84 vs
72 — yes, just about.) Which two decades look almost the same in height?
Difference questions — How much did the population grow between 1961 and 1981? (68 −
44 = 24 crore)
Trend questions — Is the graph rising or falling? Are the steps getting bigger or smaller as
we move right?
Prediction questions — Using the pattern, guess the population in 2011. (About 120 crore
— the actual figure was close to 121 crore.)
Scale questions — Why did we take 1 unit = 10 crore instead of 1 unit = 1 person?

Tip: a question like “Why did the population increase?” cannot be answered from the
graph — the graph shows what happened, not why. Such questions need other
information.

Page 30 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q2 How much did the population of India increase over 50 years? How much did the
population increase in each decade?

ANSWER

Over the 50 years from 1951 to 2001:

102 crore − 36 crore = 66 crore

That is, India added about 66 crore people in 50 years — nearly three times its 1951 population
became the 2001 population.
Decade by decade, subtract each value from the next:

DECADE POPULATION AT START POPULATION AT END INCREASE (CRORE)

1951 – 1961 36 44 8

1961 – 1971 44 54 10

1971 – 1981 54 68 14

1981 – 1991 68 84 16

1991 – 2001 84 102 18

Total — — 66

8 + 10 + 14 + 16 + 18 = 66 crore ✔ — the decade increases add up to the 50-year increase.

What the numbers show: the increase itself grew every decade — 8, then 10, 14,
16, 18. So the graph does not rise in equal steps; each step is taller than the one
before it. That is why the bars seem to climb faster and faster towards the right.

In-text Questions — Page 93

Page 31 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Section 4.4 Drawing a Bar Graph

Q1 On which item does Imran’s family spend the most and the second most?

ANSWER

Here the scale is 1 unit length = ₹ 200, so the tallest bar means the biggest expense.

Monthly expenditure of Imran’s family
3600 3400

3200 3000

2800
Expenditure (in ₹)

2400

2000

1600
1200
1200
800
800 600
400
400

0
House Food Educa- Elec- Trans- Miscel-
rent tion tricity port laneous

Item

Imran's family budget. Food and house rent tower above everything else.

Food ₹ 3400 > House rent ₹ 3000 > Miscellaneous ₹ 1200 > Education ₹ 800 > Transport ₹

600 > Electricity ₹ 400

The family spends the most on food (₹ 3400) and the second most on house rent (₹ 3000).

Total monthly expenditure = 3000 + 3400 + 800 + 400 + 600 + 1200 = ₹ 9400

Food and rent together = 3400 + 3000 = ₹ 6400 — that is nearly two-thirds of the whole

budget.

Page 32 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q2 Is the cost of electricity about one-half the cost of education?

ANSWER

Yes — it is exactly one-half.

Cost of education = ₹ 800

Half of ₹ 800 = 800 ÷ 2 = ₹ 400

Cost of electricity = ₹ 400 ✔

The bars say the same thing without any arithmetic: the electricity bar is 2 unit lengths tall and
the education bar is 4 unit lengths tall — one is exactly half of the other.

Why bar graphs make “half” and “double” easy to see: since every bar starts from
the same zero line and has the same width, one bar being half as tall really does
mean half the amount. If the bars did not start from zero, this comparison would be
false.

Q3 Is the cost of education less than one-fourth the cost of food?

ANSWER

Yes.

Cost of food = ₹ 3400

One-fourth of ₹ 3400 = 3400 ÷ 4 = ₹ 850

Cost of education = ₹ 800

800 < 850, so education is less than one-fourth of food ✔

It is only ₹ 50 short, so on the graph the education bar looks very close to a quarter of the food
bar — close enough that you must check with numbers before deciding.

Check it yourself the other way round: 4 × 800 = 3200, and 3200 < 3400. Four
times the education cost is still less than the food cost, which says the same thing.

Page 33 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
m.
Figure it Out — Pages 93 – 99
m ase
.co
Section 4.4 Drawing a Bar Graph
a g l
a s em
aQ1gl Samantha visited a tea garden, and collected data of the insects and critters she
saw there. Here is the data she collected — Mites 6, Caterpillars 10, Beetles 5,

com
Butterflies 3, Grasshoppers 2. Help her prepare a bar graph representing this data.

e m . ag
g l as
a
ANSWER

The largest number is only 10, so the simplest scale works: 1 unit length = 1 insect.

1. Draw a horizontal line and a vertical line meeting at 0.
. c om
2. On the horizontal line mark the five kinds of creatures, equally spaced. em
o
3. On thecvertical g as
m line mark 0, 2, 4, 6, 8, 10 — equal gaps for equal numbers.
l
m . a
a s e
l
4. Draw bars of the same width with the same gap between them: heights 6, 10, 5, 3 and 2
ag units.
s
5. Give the graph a title and label both lines.
m a
m .co agl
l a se
g
Insects and critters seen in the tea garden
a
12

co m
.
10

m
10

m as e
.co 8 a g l
se m
Number seen

g l a
a
6
6

m
5

a se
com g l
m. a
4
3

ase
agl
2
2

m Grass-
0
Mites Cater- Beetles Butter-
. co
e m
las
pillars flies hoppers

co m
e m . ag
as
Insects and critters

a g l
.c
Samantha's bar graph. Caterpillars were seen the most (10) and grasshoppers the least (2).

s e m
com a
Total creatures seen = 6 + 10 e+m
. agl
l a s 5 + 3 + 2 = 26

10g — nearly two out of every five creatures she saw.
Caterpillars alone are a

co m
m .
m ase
.co


a g l Page 34 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Tip: a bar graph must always start from 0. If we started the vertical line at 2, the
grasshopper bar would vanish and the beetle bar would look three times smaller
than it really is.

Q2 Pooja collected data on the number of tickets sold at the Bhopal railway station for
a few different cities of Madhya Pradesh over a two-hour period — Vidisha 24,
Jabalpur 20, Seoni 16, Indore 28, Sagar 16. She used this data and prepared a bar
graph on the board to discuss the data with her students, but someone erased a
portion of the graph. a. Write the number of tickets sold for Vidisha above the bar.
b. Write the number of tickets sold for Jabalpur above the bar. c. The bar for Vidisha
is 6 unit lengths and the bar for Jabalpur is 5 unit lengths. What is the scale for this
graph? d. Draw the correct bar for Sagar. e. Add the scale of the bar graph by
placing the correct numbers on the vertical axis. f. Are the bars for Seoni and Indore
correct in this graph? If not, draw the correct bar(s).
No. of Tickets

Vidisha Jabalpur Seoni Indore Sagar

City

Pooja's bar graph as it was left on the board — the vertical scale and one bar are
missing.

ANSWER

a. Vidisha → 24 tickets (write 24 just above that bar).
b. Jabalpur → 20 tickets.
c. The scale. The same bar is described in two ways — 6 unit lengths, and 24 tickets. So one unit
length must stand for

Page 35 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

24 tickets ÷ 6 units = 4 tickets per unit length

Check with Jabalpur: 5 units × 4 = 20 ✔

Scale: 1 unit length = 4 tickets

d. The bar for Sagar.

16 ÷ 4 = 4 unit lengths — the same height as Seoni's bar.

e. The numbers on the vertical axis must go up in steps of 4, starting from 0:

0, 4, 8, 12, 16, 20, 24, 28

f. Seoni is 16 tickets = 16 ÷ 4 = 4 units; Indore is 28 tickets = 28 ÷ 4 = 7 units.

Seoni's bar is correct — on the board it is drawn 4 unit lengths tall, exactly as it should be.
Indore's bar is wrong. On the board it is drawn only about 4½ unit lengths — barely taller
than Seoni's, and shorter than Vidisha's. Indore sold the most tickets, so its bar must be 7
unit lengths and must be the tallest bar of all.

The corrected graph is shown below.

Tickets sold at Bhopal station in two hours
28
28
24
24
20
20
No. of Tickets

16 16
16

12

8

4

0
Vidisha Jabalpur Seoni Indore Sagar

City

The corrected bar graph. Indore (28) is now the tallest bar; Seoni and Sagar are equal at 16.

Page 36 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Total tickets sold in the two hours = 24 + 20 + 16 + 28 + 16 = 104

Why the scale had to be found first: without a scale a bar is just a rectangle — it
means nothing. “6 units” becomes “24 tickets” only after we know that one unit is 4
tickets. That is why the scale must always be written on a graph.

Q3 Chinu listed the various means of transport that passed across the road in front of
his house from 9 a.m. to 10 a.m. a. Prepare a frequency distribution table for the
data. b. Which means of transport was used the most? c. If you were there to collect
this data, how could you do it? Write the steps or process.

ANSWER

a. Read Chinu's list row by row and put one tally mark for each vehicle.

MEANS OF TRANSPORT TALLY MARKS NUMBER

Bike |||| |||| ||| 13

Scooter |||| |||| 9

Bicycle |||| ||| 8

Auto rickshaw |||| ||| 8

Car |||| | 6

Bus |||| 4

Bullock cart || 2

Total — 50

13 + 9 + 8 + 8 + 6 + 4 + 2 = 50 — and Chinu's list has 50 entries, so nothing has been

missed.

Page 37 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Vehicles passing Chinu’s house (9 a.m. – 10 a.m.)

Bike 13

Scooter 9

Bicycle 8

Auto rickshaw 8

Car 6

Bus 4

Bullock cart 2

0 2 4 6 8 10 12 14
Number of vehicles

The same table as a horizontal bar graph — the order of the vehicles becomes obvious at once.

b. The bike was used the most — 13 out of 50 vehicles, more than a quarter of all the traffic.
Two-wheelers together (bike 13 + scooter 9 + bicycle 8) make 30 of the 50 vehicles.
c. How I would collect this data:

1. Decide the place and the exact time first — one spot on the road, from 9:00 to 10:00 a.m.
2. Make a table beforehand with one row for each kind of vehicle, and keep a spare row for
“any other”.
3. Sit facing the road and put one tally mark for every vehicle as it passes, in bundles of five.
Do not try to write full words — vehicles pass too fast.
4. Count vehicles going in both directions (or fix one direction and say so), and never count the
same vehicle twice.
5. At the end of the hour, count the tally marks, write the frequencies, and add them up to
check the total.

Tip: two friends counting together is safer than one — one can call out and the
other can mark. During a busy hour a single person can easily miss vehicles.

Page 38 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
m.
Roll a die 30 times and record the number you obtain each time. Prepare a
se
Q4

o m l a
frequency distribution table using tally marks. Find the number that appeared: a.
g c. Find numbers
m .c
The minimum number of times. b. The maximum number of a
times.

l a se appeared an equal number of times.
that
ag
m
ANSWER

. co ag
m
This is an experiment — roll the die yourself. Here is one set of 30 actual rolls:

ase
a g l
5, 2, 6, 3, 5, 1, 4, 2, 5, 3, 6, 2, 5, 1, 3, 6, 2, 5, 4, 1, 3, 5, 2, 6, 1, 3, 2, 5, 4, 6

. com
a s em
com
NUMBER ON THE DIE TALLY MARKS FREQUENCY

m . agl
ase
1 |||| 4

agl2 |||| | 6

m a s
.co agl
3 |||| 5

sem |||
4
g l a
a
3

5 |||| || 7

co m
m .
se
6 |||| 5

com l a
m . ag 30
ase
Total —

agl For this experiment:
se m
coofmtimes — 7 times. l a
a. 4 appeared the minimum number of times — only 3 times.
. a g
em of times — 5 times each.
b. 5 appeared the maximum number
s
c. 3 and 6 appeared an equalanumber
agl
m
Why your answers will be different: a die has no memory and no favourite face.
Every roll is fresh, so a different set of 30 rolls gives different counts. Yet the
. co
se m
o m l a
frequencies stay near 5 each, because 30 ÷ 6 = 5. Roll the die 600 times and every
agthe more even the pattern
.cface will come close to 100 — the more data you collect,
m becomes.
l a se
ag
.c
s e m
m a
co agl
Check it yourself: the frequencies must always add up to 30. If they do not, a roll
.
s em
has been missed while marking.
a
agl

com
m .
m ase
.co


a g l Page 39 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q5 Faiz prepared a frequency distribution table of data on the number of wickets taken
by Jaspreet Bumrah in his last 30 matches. a. What information is this table giving?
b. What may be the title of this table? c. What caught your attention in this table? d.
In how many matches has Bumrah taken 4 wickets? e. Mayank says, “If we want to
know the total number of wickets he has taken in his last 30 matches, we have to
add the numbers 0, 1, 2, 3 …, up to 7.” Can Mayank get the total number of wickets
taken in this way? Why? f. How would you correctly figure out the total number of
wickets taken by Bumrah in his last 30 matches, using this table?

WICKETS TAKEN NUMBER OF MATCHES

0 2

1 4

2 6

3 8

4 3

5 5

6 1

7 1

The frequency distribution table Faiz prepared.

ANSWER

a. The table tells us how many matches Bumrah took each number of wickets in. For example,
the row “2 — 6” means that in 6 of the matches he took exactly 2 wickets. It does not tell us
which match was which, or against whom he played.
b. A good title would be “Wickets taken by Jaspreet Bumrah in his last 30 matches”. (Other
titles work too — “Bumrah's wicket-taking record: last 30 matches”.) A title must say whose data
it is, what is being counted and over what period.
c. Things worth noticing:

Taking 3 wickets is his most usual performance — it happened in 8 matches, more than any
other.
He went wicketless in only 2 matches out of 30.
A haul of 6 or 7 wickets is rare — just 1 match each.

Page 40 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

The frequencies rise up to 3 wickets and then fall — but 5 wickets (5 matches) breaks the fall.
The numbers of matches add up to 2 + 4 + 6 + 8 + 3 + 5 + 1 + 1 = 30 ✔, which is exactly what
the title promises.

Wickets taken by Bumrah in his last 30 matches
8
8

7
6
6
Number of matches

5
5
4
4
3
3
2
2
1 1
1

0
0 1 2 3 4 5 6 7

Wickets taken

Faiz's table drawn as a bar graph. The tallest bar is at 3 wickets.

d. Bumrah took 4 wickets in 3 matches.
e. No, Mayank's method is wrong.

Mayank would get 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

That would be the answer only if he had played exactly one match of each kind. But he took 3
wickets in 8 different matches, so those 3 wickets must be counted 8 times, not once. Mayank is
adding the labels of the rows instead of using the counts.
f. The correct method: multiply each number of wickets by the number of matches, then add.

Page 41 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

WICKETS TAKEN NUMBER OF MATCHES WICKETS IN THOSE MATCHES

0 2 0×2=0

1 4 1×4=4

2 6 2 × 6 = 12

3 8 3 × 8 = 24

4 3 4 × 3 = 12

5 5 5 × 5 = 25

6 1 6×1=6

7 1 7×1=7

Total 30 matches 90 wickets

0 + 4 + 12 + 24 + 12 + 25 + 6 + 7 = 90 wickets

Did you know? 90 wickets in 30 matches means an average of 90 ÷ 30 = 3 wickets
per match — an outstanding record for a fast bowler.

Page 42 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q6 The following pictograph shows the number of tractors in five different villages.
Observe the pictograph and answer the following questions— a. Which village has
the smallest number of tractors? b. Which village has the most tractors? c. How
many more tractors does Village C have than Village B? d. Komal says, “Village D has
half the number of tractors as Village E.” Is she right?

VILLAGES NUMBER OF TRACTORS (

= 1 TRACTOR )

Village A

Village B

Village C

Village D

Village E

The pictograph given in the book.

Page 43 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
m.
ANSWER

m as e
.co
The key is “one symbol = 1 tractor”, so count the symbols in each row.
a g l
se m
g l a
a Number of tractors in five villages
= 1 tractor

co m
Village A
e m . 6 ag
g l as
a
Village B 5

Village C 8

co m
m.
Village D 3

m as e
.co
Village E 6
a g l
se m
g l a
a
The tractor pictograph. Village C has the longest row and Village D the shortest.

m a s
VILLAGE A
m .co B C D E
agl
l a se
Tractors
a g 6 5 8 3 6

a. Village D — only 3 tractors, the shortest row.
co m
m .
e
b. Village C — 8 tractors, the longest row.
m l as
.co a g
c. Village C has

se m
g l a
a 8 − 5 = 3 more tractors than Village B

se m
com g l a
d. Yes, Komal is right.
m . a
ase
Village E has 6 tractors agl

co m
.
Half of 6 = 6 ÷ 2 = 3

e m
m l as
Village D has 3 tractors ✔

m .co a g
l a se
ag
c
Check it yourself: the five villages have 6 + 5 + 8 + 3 + 6 = 28 tractors altogether.
m .
m a s e
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 44 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q7 The number of girl students in each class of a school is depicted by the pictograph
(one symbol = 4 girls). Observe this pictograph and answer the following questions:
a. Which class has the least number of girl students? b. What is the difference
between the number of girls in Classes 5 and 6? c. If two more girls were admitted
in Class 2, how would the graph change? d. How many girls are there in Class 7?

Page 45 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

CLASSES NUMBER OF GIRL STUDENTS (

= 4 GIRLS )

1

2

3

4

Page 46 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

5

6

7

8

The pictograph given in the book.

ANSWER

Here one symbol stands for 4 girls, so a half symbol stands for 2 girls. Every row must first be
turned into a number:

number of girls = (number of symbols) × 4

Page 47 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Number of girl students in each class
= 4 girls

Class 1 24

Class 2 18

Class 3 20

Class 4 14

Class 5 10

Class 6 16

Class 7 12

Class 8 6

The pictograph read row by row. Class 1 has the longest row (6 symbols) and Class 8 the shortest (1½
symbols).

CLASS 1 2 3 4 5 6 7 8

Symbols 6 4½ 5 3½ 2½ 4 3 1½

Girl students 24 18 20 14 10 16 12 6

a. Class 8 — its row is the shortest of all eight rows: 1½ symbols, that is 1 × 4 + 2 = 6 girls, the
least in the school.
b. Class 5 shows 2½ symbols and Class 6 shows 4 symbols:

Class 5 = 2½ × 4 = 2 × 4 + 2 = 10 girls

Class 6 = 4 × 4 = 16 girls

Difference = 16 − 10 = 6 girls

c. Class 2's row is 4½ symbols, that is 4 × 4 + 2 = 18 girls. With 2 more girls it becomes

18 + 2 = 20 girls = 20 ÷ 4 = 5 full symbols

So the half symbol at the end of Class 2's row would become a full symbol, and the row
would then show 5 complete symbols. Nothing else in the graph changes.
d. Class 7 shows 3 symbols:

Page 48 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
e m.
3 × 4 = 12 girls
m l as
m .co a g
l a se
g
Why the scale matters so much here: the row lengths tell you at once which class
ahas more girls, but they do not give the numbers until you multiply by 4. A reader
m
who forgets the key would say Class 7 has “3 girls” instead of 12 — a mistake of four

. co ag
m
times.

ase
a g l
Check it yourself: the school has 24 + 18 + 20 + 14 + 10 + 16 + 12 + 6 = 120 girl
students in all, that is 30 symbols on the pictograph.
co m
em.
m l as
.co a g
a s em
l
Mudhol Hounds (a type of breed of Indian dogs) are largely found in North
g
Q8
a Karnataka’s Bagalkote and Vijaypura districts. The government took an initiative to

s
protect this breed by providing support to those who adopted these dogs. Due to

om a
. c agl
this initiative, the number of these dogs increased. The number of Mudhol dogs in
six villages of Karnataka are asm
s e follows — Village A : 18, Village B : 36, Village C : 12,

a g laVillage F : 24. Prepare a pictograph and answer the
Village D : 48, Village E : 18,
following questions: a. What will be a useful scale or key to draw this pictograph? b.

m
How many symbols will you use to represent the dogs in Village B? c. Kamini said

. co
m
that the number of these dogs in Village B and Village D together will be more than

as e
om l
the number of these dogs in the other 4 villages. Is she right? Give reasons for your
.cresponse. a g
se m
g l a
a ANSWER

se m
m— every one of them is a multiple of 6.
a. Look for a number that divides all six values exactly, and that is big enough to keep the rows
co24 g l a
. a
em
short. The numbers are 18, 36, 12, 48, 18,

a s
agl
18 ÷ 6 = 3 36 ÷ 6 = 6 12 ÷ 6 = 2 48 ÷ 6 = 8 18 ÷ 6 = 3 24 ÷ 6 = 4

co m
m .
as e
A useful key is “one symbol = 6 dogs”. Every row then comes out in whole symbols — no half
m
.co g l
or quarter pictures are needed at all. (A key of 12 would give 1½, 3, 1, 4, 1½, 2 — it works but
a
a s em needs half symbols; a key of 2 or 3 would make the rows uncomfortably long.)

agl
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 49 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Mudhol hounds in six villages of Karnataka
= 6 dogs

Village A 18

Village B 36

Village C 12

Village D 48

Village E 18

Village F 24

The pictograph drawn with the key “1 symbol = 6 dogs”. Village D has the longest row, Village C the
shortest.

b. Village B has 36 dogs, so

36 ÷ 6 = 6 symbols

c. Yes, Kamini is right.

Villages B and D = 36 + 48 = 84 dogs

Other four villages (A, C, E, F) = 18 + 12 + 18 + 24 = 72 dogs

84 > 72, so Kamini is correct — by 12 dogs

The picture shows it too: the rows of B and D together have 6 + 8 = 14 symbols, while the other
four rows have 3 + 2 + 3 + 4 = 12 symbols.

Check it yourself: all six villages have 84 + 72 = 156 dogs, which is 156 ÷ 6 = 26
symbols in the whole pictograph.

Page 50 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q9 A survey of 120 school students was conducted to find out which activity they
preferred to do in their free time — Playing 45, Reading story books 30, Watching TV
20, Listening to music 10, Painting 15. Draw a bar graph to illustrate the above data
taking the scale of 1 unit length = 5 students. Which activity is preferred by most
students other than playing?

ANSWER

First turn every frequency into a bar height, using the given scale 1 unit length = 5 students:

PREFERRED ACTIVITY NUMBER OF STUDENTS HEIGHT OF BAR

Playing 45 45 ÷ 5 = 9 units

Reading story books 30 30 ÷ 5 = 6 units

Watching TV 20 20 ÷ 5 = 4 units

Listening to music 10 10 ÷ 5 = 2 units

Painting 15 15 ÷ 5 = 3 units

Total 120 24 units

Now mark the vertical line 0, 5, 10, 15, … 50 and draw bars of equal width with equal gaps.

Free-time activity preferred by 120 students
50
45
45
40
Number of students

35
30
30
25
20
20
15
15
10
10
5
0
Playing Reading Watching Listening Painting
story books TV to music

Preferred Activity

Page 51 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

The survey drawn as a bar graph. Playing towers over everything; reading story books comes next.

Other than playing, reading story books is preferred by the most students — 30 out of 120.

Check: 45 + 30 + 20 + 10 + 15 = 120 ✔ (all the students surveyed are accounted for)

Why this scale is a good choice: every frequency is a multiple of 5, so every bar
ends exactly on a marked line — no bar has to stop in between two lines. And 120
students fit in only 9 units of height, so the graph is small enough for a notebook
page.

Page 52 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q10 Students and teachers of a primary school decided to plant tree saplings in the
school campus and in the surrounding village during the first week of July. Details
of the saplings they planted are given in the bar graph. a. The total number of
saplings planted on Wednesday and Thursday is ___________. b. The total number of
saplings planted during the whole week is ___________. c. The greatest number of
saplings were planted on ___________ and the least number of saplings were planted
on ___________. Why do you think that is the case? Why were more saplings planted
on certain days of the week and less on others? Can you think of possible
explanations or reasons? How could you try and figure out whether your
explanations are correct?

70

60
Number of saplings planted

50

40

30

20

10

0
Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Day

The bar graph given in the book.

ANSWER

Read the height of each bar against the vertical line, where the marks go up in tens.

Page 53 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
em.
as
Saplings planted in the first week of July
m l
70
.co a g
a s em60
gl
60

a
Number of saplings planted

52
50
50
40
. com40 40
ag
40
s30em
l a
30 ag
20
co m
e m.
m l as
10

.co a g
a s em 0

ag l Monday Tuesday Wednes-
day
Thursday Friday Saturday Sunday

m a s
.co agl
Day

se m
g l a
Saplings planted on each day of the week. Saturday's bar is the tallest and Wednesday's the shortest;
a
Monday's bar stops a little above the 50 line.

. com
m
ase
DAY MON TUE WED THU FRI SAT SUN
m
m . co ag50l
se
Saplings 52 40 30 40 60 40

l a
ag Six of the bars end exactly on a marked line. Monday's bar is the only one that does not — it
rises a little above the 50 line, at about 52.
se m
a. Wednesday and Thursday together: com g l a
m . a
as e
g l
30 + 40 = 70 saplings a

co m
m .
e
b. The whole week:
m l as
.co
m 52 + 40 + 30 + 40 + 50 + 60 + 40 a g
l a se
ag = 52 + (40 + 40 + 40) + 30 + 50 + 60
.c
s e m
m a
co agl
= 52 + 120 + 140 = 312 saplings
m .
ase
a g l

co m
m .
m ase
.co


a g l Page 54 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Note: the answer key at the back of some books gives 310 for this part. That
happens if Monday's bar is read as 50. Look carefully at the graph — Monday's bar
clearly rises above the 50 line, so the correct total is 312. (If your teacher reads
Monday as 50, the total becomes 310.)

c. The greatest number of saplings were planted on Saturday (60) and the least on Wednesday
(30).
Possible reasons:

Saturday is usually a half day or a holiday, so more students, teachers and even parents
were free to join the plantation drive.
Wednesday was a full working day with regular classes, so only a short time was left for
planting. It may also have rained heavily, or the supply of saplings may have run short that
day.
July is the sowing month — soft, wet soil makes planting easy, which is why the numbers are
large on every day.

How to check whether these explanations are right:

Ask the teachers and students how much time was given for planting on each day, and note
it beside the graph.
Look at the school attendance register — were fewer people present on Wednesday?
Check the rainfall record for that week, and the record of how many saplings were delivered
each day.
Repeat the drive next month and see whether Saturday is again the biggest day. If the same
pattern appears again, the explanation is much more believable.

Remember: a bar graph shows what happened. The reason is never inside the graph
— it has to be found by collecting more information.

Page 55 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Q11 The number of tigers in India went down drastically between 1900 and 1970.
Project Tiger was launched in 1973 to track and protect the tigers in India. Starting
in 2006, the exact number of tigers in India was tracked. Shagufta and Divya
looked up information about the number of tigers in India between 2006 and 2022
in four-year intervals. They prepared a frequency table for this data and a bar
graph to present this data, but there are a few mistakes in the graph. Can you find
those mistakes and fix them?

YEAR NUMBER OF TIGERS (APPROX.)

2006 1400

2010 1700

2014 2200

2018 3000

2022 3700

Number of Tigers in India

2022

2018
Year

2014

2010

2006

0 1000 2000 3000 4000

Number of Tigers

Shagufta and Divya's frequency table, and the bar graph they drew from it.

ANSWER

The horizontal line is marked 0, 1000, 2000, 3000, 4000, so 1 unit length = 1000 tigers. Read
each bar against that scale and compare it with the table.

Page 56 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

YEAR TABLE SAYS THE GRAPH SHOWS CORRECT?

2006 1400 about 800 — the bar stops well before the 1000 mark ✘ too short
2010 1700 about 1500 — it stops halfway between 1000 and 2000 ✘ too short
2014 2200 about 2900 — nearly at the 3000 mark ✘ too long
2018 3000 about 2200 — just past the 2000 mark ✘ too short
2022 3700 about 3700 ✔ correct

The mistakes: only the 2022 bar is drawn correctly. Of the other four:

The 2014 and 2018 bars have been interchanged — 2014 has been given the 2018 value
(about 3000) and 2018 has been given the 2014 value (about 2200). This is easy to spot
without any measuring: the 2018 bar comes out shorter than the 2014 bar, yet the table says
the tigers increased from 2200 to 3000 in those years.
The 2006 bar is far too short — about 800 instead of 1400.
The 2010 bar is too short — about 1500 instead of 1700.

How to fix them: redraw the four wrong bars to the lengths the table gives — 2006 to 1400 (a
little less than 1½ units), 2010 to 1700, 2014 to 2200 and 2018 to exactly 3000 (3 units). Keep
every bar starting from 0, of the same width and with equal gaps. After the fix the bars grow
steadily longer from 2006 to 2022, as they must.

Number of Tigers in India (corrected)

2022 3700

2018 3000

2014 2200

2010 1700

2006 1400

0 1000 2000 3000 4000
Number of Tigers

The corrected bar graph. Each bar now matches the table, and the steady rise since 2006 is clear.

Growth from 2006 to 2022 = 3700 − 1400 = 2300 tigers

That is more than two and a half times the 2006 count, in just 16 years.

Page 57 of 65

Class 6 Maths Chapter 4 Data Handling and Presentation AglaSem · NCERT Solutions

Did you know? India is home to about three-quarters of all the wild tigers in the
world. Project Tiger, started in 1973, is one of the most successful animal-protection
programmes anywhere.

In-text Questions — Page 101
Section 4.5 Artistic and Aesthetic Considerations

Q1 How much taller is Mount Everest than Mount Koscuiszko? Are Mount Denali and
Mount Kilimanjaro very different in height?

ANSWER

Everest against Koscuiszko:

8848 m − 2228 m = 6620 m

Mount Everest is 6620 metres taller than Mount Koscuiszko — that is, Everest is almost four
times as tall (4 × 2228 = 8912, only a little more than 8848).
Denali against Kilimanjaro:

6194 m − 5895 m = 299 m

No, they are not very different. The gap of 299 m is only about one-twentieth of either
mountain's height, so on a bar graph their bars look almost equal.

Page 58 of 65

as e
Class 6 Maths Chapter 4 Data Handling and Presentation
a g l AglaSem · NCERT Solutions

co m
e m.
as
Tallest mountain on each continent
m l
m .co Asia — Everest a g
ase
8848

l America — Aconcagua
agSouth 6962

North America — Denali 6194

co m
Africa — Kilimanjaro
e m . 5895
ag
g l as
a
Europe — Elbrus 5642

Antarctica — Vinson Massif 4892

co m
m.
Australia — Koscuiszko 2228

m l a se 8000 9000
.co ag
0 1000 2000 3000 4000 5000 6000 7000

m
ase
Height (in metres)

agl The seven tallest mountains as horizontal bars. Denali and Kilimanjaro are nearly the same length;
Koscuiszko is far shorter than the rest.

m a s
m .co agl
l a se
g
Why the bar graph beats the table: from a table of seven numbers you must
a
subtract in your head before you can compare. In the picture the eye does the work

m
— the Denali and Kilimanjaro bars end almost at the same place, while the
Koscuiszko bar is obviously the odd one out.
. co
em
m l as
.co a g
a s em
a l
gFigure it Out — Page 103
Section 4.5 Artistic and Aesthetic Considerations
se m
com g l a
m . a
ase
agl
Q1 If you wanted to visually represent the data of the heights of the tallest persons in
each class in your school, would you use a graph with vertical bars or horizontal
bars? Why?

co m
m .
as e
com bars (a column graph). l
ANSWER

. a g
sem
Vertical

a
agl c
.
Why: a person's height is measured upward from the ground. A vertical bar also

s e m
m a
grows upward from the base line, so the picture works the same way as the thing it

m . co
stands for. The reader compares the tops of the columns exactly as they would
e agl
l as
compare children standing side by side against a wall — and understands it without
g
being told. a

co m
m .
m as e
.co


a g l Page 59 of 65