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NCERT 6TH CLASS MATHEMATICS CHAPTER DATA HANDLING

INTRODUCTION

 
Date :
Data is the collection of information or facts. It may be a collection of numbers, figures, words or symbols.
 
Raw Data :
The data that has been collected in the original form is called raw data.
 
For example :
The marks obtained by 50 students in the Social Studies exam are 60, 40, 35, 27, 67, 72, 40, 37, 45, 50, 52 and so on up to 50 items.
This collection of data is raw data.
 
Observation :
Each of the items in the raw data is called an observation. The marks of each student stands for an observation in the raw data.
 
Type of Data :
· There are two types of data — primary data and secondary data.
· Primary data is the data collected directly from the source.
· Secondary data is the data collected from secondary sources such as the internet, TV, libraries, 
   newspapers. etc
 
For example :
Mr Sen asks the students to give their choice for the particular location they would like to go, the games they would like to play, the type of food they would like to have on their trip, he is getting the information directly from the source. This is primary data collection. When he is collecting information about weather, or accommodation facilities. he may collect these from the internet, travel brochures, newspapers, magazines, television and other souces. These sources are secondary sources is called secondary data.
 
Organisation of Data :
To draw a meaningful conclusion from the data, the data must be organized. Understanding and analyzing the data becomes easier when it is represented by pictures or tables.
 
Tabular form :
Consider the data given here 75%, 88%, Devan, Charu, Sajita, Arvind, 70%, 60%, 83%, 90%, 94%, Greg, Manisha, Vijayan, Ahmed, Raghu, Neetu, 66%, 77%, 72%. This is raw data.
The data given above are facts in words and figures. Unless the available data is property presented, we cannot get meaningful information from it. Given below is a tabular representation of the above data. This makes the information easy to understand.
 
 Frequency distribution :
Let us study the peak hour traffic at a road junction to share it with the town or city addministrators to help them develop that junction for easy traffic flow. We need to collect the data about the number of trucks, cars, other motor vehicles, and cycles passing through that junction during a fixed period at the peak hour.
We will try to find the number of vehicles passing through that junction during the peak interval of 9.00 a.m. to 9.30 a.m.
For this you have to prepare a data collection sheet.
 
 
You find here two new words tally and frequency. Tally is a mark 'l' we put in the chart against each item. For example, if a car passes the junction, we mark one tally 'l' in the box against car. When four cars have passed, the tally will look like this "llll'. When the fifth car passes, we do not write 'lllll' but we put a cross across the four tallies like . This forms a group of 5. Similarly, '  ||' will be two fives and two, i.e., 12. The total value of the tallies in each row tells us the number of the particular type of vehicle which passed through that junction. In general, the tally mark represents the number of times that particular situation happened. This is called the frequrency of that event or that happening. In the  case of trucks, 17 trucks passed through that junction between 9.00 a.m. and 9.30 a.m. Thus the frequency of the trucks passing during 9.00—9.30 is 17. And the  frequency for cars is 33,
...
so on. Finally the investigation or data collection chart will look like this.
 
Representation of Data :
We have learnt how to collect the information and how to organize it using a frequency table. To represent the study of this data, we can draw graphs for this data. Pictographs and bar graphs are simple ways of representing this data.
 
PICTOGRAPH
A pictograph is the way of representing data using pcitures.
In a school compound, there are 5 Neem trees, 4 Coconut trees and 3 Ashoka trees.
Let us represent the above data by a pictograph.
Trees in the school compound
 
 
Let us now suppose that there are 40 neem trees, 24 coconut trees and 32 ashoka trees, you cannot draw each picture for each tree. So we take a convenient scale. Let us consider that each tree drawn represents 8 actual trees. 40 neem trees well become 5 trees drawn. Similarly, 24 coconut trees will be 3 and 32 ashoka trees will be represented by 4 trees. Now it is convenient to draw a graph.
 
 
Make sure that each picture must be one below the other.
Reading and intrepreting pictograph :
Reading a pictograph helps in intrepreting the pictograph so as to get information from it.
Ex. Given below is a pictograph of ages of students in a class of 51 students.
Key : · represents 3 students
 
 

Answer the following questions.

(i) What is the total number of students ?
Sol. · represent 3 students and here we have 17 such
· \ Total no. of students = 17 × 3 = 51
 
(ii) Which age group has the lowest number of students ? How many students are there in group.
Sol. 65 month old age group has the lowest number of students. Three students are there in group.
 
(iii) Which age group has the highest number of students ?
How many students are there in the group ?
Sol. 62 months old age group has the highest number of students. 15 students are there in the group.
 
(iv) How many students are there, who are 62 months and below ?
Sol. · represents 3 students and here we have 10 such
· who are 62 months and below
\ Total no. of students who are 62 months & below = 10 × 3 = 30
 
(v) How many students are 62 months and above ?
Sol. · represents 3 students and here we have 12 such 
· who are 62 months and above
\ Total no. of students who are 62 months & above = 12 × 3 = 36
 
(vi) How many students are between the age of 61 months and 64 months ?
Sol. · represent 3 students and here we have 9 such · who are between the age of 61 months and 64 months
\ Total no. of students who are between the age of 61 months and 64 months = 9 × 3 = 27.
 

BAR GRAPH

When the given data is large and also not in multiples of a number, it is difficult to draw pictographs. When numerial data is presented as columns on a graph, this graphical representation of data is called a bar graph. It is also known as column graph.
 

Properties of Bar Graphs :

In a bar graph, the columns are drawn such that :
(a) Each column is of equal width.
(b) There is equal space between each column.
(c) The height of each column indicates the value of data such that if a perpendicular is drawn from its top to ray OY, the percentage of marks represented by the column can be read off.
(d) All bars or columns are draw on the same common base.
(e) To make the bar graph attractive, the bars may be coloured.
(f) Bars are very helpful for visual comparison of data.
 

Construction of a bar graph :

STEPS :

1. On the graph paper draw a horizontal line (x-axis) and a vertical line (y-axis)
2. Along the horizontal line, mark points at equal distances and write the names of the items for which the data is to be represented.
3. Choose a suitable scale which shall determine the height of the bars.
4. Locate the heights of different bars according to scale.
5. On x-axis draw bars of equal width according to the required height.
6. Keep the distance between the bars the same.
Ex. Look at the frequency chart representing the number of tourists staying in Hotal Raj Mahal in Shimla for 5 months : March, April, May, June and July.
Months                  Tourists
March                     120
April                        180
May                         240
June                        210
July                         120
Take a graph paper or a squared paper. Draw a horizontal axis at the bottom. On the lift side draw a vertical axis. Take an appropriate scale. In this case, let it be 1 division (1 centimetre) equal to 15 tourists.
 
 
On the horizontal axis, represent the five months separately with equal width and equal distances between them.
Reading of bar graphs :
Reading of the bar graph is the process of interpreting the bar graph to gather information.
Ex. Read the bar graph given below and answer the following questions.
 
 
(i) What is the total number of students in all the clubs together ?
Sol. 20 + 30 + 40 + 20 = 110 students
 
(ii) Which club has more students and how many ?
Sol. Drama club, 40 students.
 
(iii) Which club have equal number of students ?
Sol. Cricket club and music club.
 
(iv) How many students are there in hockey club ?
Sol. 30 students.
 
(v) What information is given in the bar graph ?
Sol. The bar graph represents the number of members of different sports clubs.
 

AVERAGE

Average =
                  
Ex. Find the average of all prime numbers between 30 and 50.
Sol. There are five prime numbers between 30 and 50.
They are 31, 37, 41, 43 and 47
\ Required average = (31+37+41+43+47)/5
= 199/5
= 39.8
 
Ex. Find the average of first 5 multiple of 7.
Sol. Required average = (7+14+21+28+35)/5
= 105/5
= 21

OBJECTIVE TYPE


Q.1 If one symbol  represents 150 trees, then the number of trees represented by  are
(A) 600 (B) 700 (C) 750 (D) 500
Direction : (Q. 2 to Q. 5)
Using the given bar graph, choose the correct option in the questions given below :

Q.2 Which colour is the least colour common among the cars in the parking lot ?
(A) black                          (B) white                          (C) red                           (D) grey


Q.3 Which colours had the same number of cars parked ?
(A) Black and Grey         (B) White and Black         (C) White and Red        (D) Grey and White


Q.4 What is the ratio of the number of white cars to red cars ?
(A) 5 : 2                           (B) 2 : 5                             (C) 3 : 1                        (D) 1 : 3


Q.5 The total number of cars in the parking lot was
(A) 20                              (B) 35                             (C) 50                               (D) 40


Q.6 The picture graph shows the capacity of 4 containers.
Which container has a capacity of 1/2 litre ?

(A) Bottle                    (B) Flask                 (C) Mug                   (D) Glass
Q.7 Total numbers of animals in five villages are shown in the following table.

If one symbol Å represents 10 animals, the total number of symbols required to represent animals for village C will be :
(A) 5                             (B) 7                      (C) 8                      (D) 9 


Q.8 The following pictograph shows the number of varieties of apples stored in a supermarket. The total number of apples stored in the supermarket is

(A) 150                 (B) 100                    (C) 50                       (D) 16


Q.9 Final marks of 20 students are as follows :
53, 61, 48, 60, 78, 68, 55, 100, 67, 90, 75, 88, 77, 37, 84, 58, 60, 48, 62, 56


What is the lowest score ?
(A) 37                (B) 100                 (C) 62                   (D) 48
Q.10 Find the average of first 5 prime numbers
(A) 5                  (B) 6                     (C) 7.6                  (D) 5.6
Q.11 The average of first five multiples of 2 is
(A) 3                  (B) 5                     (C) 6                     (D) 8
Q.12 The arithmetic mean of the numbers 12, 19, 13 and 16 is
(A) 12                (B) 13                   (C) 14                   (D) 15

SUBJECTIVE TYPE


Q.1 The table below shows the number of students playing four different games :

Present this information as a pictograph.


Q.2 Read the bar graph in the figure and answer the following questions :
(i) What is the information given by the bar graph ?
(ii) What is the number of families having 6 members ?
(iii) How many members per family are there in the maximum number of families ? Also tell the number of such families.
(iv) What are the number of members per family for which the number of families are equal ? Also tell the number of scuh families.

 

Q.3 Draw the bar graph of the following data.

Q.4 Draw the bar graph of the following data

Q.5 Read the graph and answer.

Fill in the chart and prepare a frequency table.

Q.6 A class took a vote to decide where to go for a trip.

(a) How many students are there in the class ?
(b) Where did the majority of students want to go ?
(c) Which was the second choice ?
(d) How many students voted for water park ?

Q.7 Make a bar graph for the following data.
(a) The table shows the number of books read by the class in the past month.

(b) This is the daily routine of Shankar. Make a bar graph representing the time and activities.
Activity             Time (hours)
School                       6
Sleeping                    9
Playing + hobby        3
TV watching              2
Eating + Drinking      1
Other activities          3
(c) This is the result of a traffic survey conducted by Anil in front of his village house for 1 hour.

(d) For the 'Marathon Run for Peace' programme a big group of children took part. The age—wise details of participants are given below.


Q.9 The music group of Class Vl of St. George's School performed a musical programme. The students of Class Vll were asked to grade the programme in a scale of 5 where 1 represents excellent performance, 2 good, 3 ordinary performance, 4 not good, 5 very poor show. This is the result of 80 students of Class Vl.
4, 2, 5, 2, 4, 5, 3, 4, 3, 2, 5, 5, 3, 2, 4, 5, 4, 5, 4, 4, 3, 3, 4, 3, 4, 3, 5, 5, 4, 3, 2, 5, 4, 4, 3, 3, 4, 4, 5, 5, 4, 3, 2, 5, 4, 4, 5, 3, 3, 4, 4, 3, 3, 4, 5, 5, 3, 2, 4, 3, 5, 4, 3, 2, 4, 2, 4, 3, 3, 4, 2, 4, 4, 3, 4, 2, 4, 4
Prepare a frequency table using tally marks.


ANSWER KEY-1

OBJECTIVE :


1. C 2. C 3. A 4. C 5. D 6. C 7. D

8. A 9. A 10. A 11. C 12. D

SUBJECTIVE :
1. (i) maximum number of families with different number of members in family
(ii) 50
(iii) 3, 120
(iv) 8, 9, 10; 10


5. Monday (10), Tuesday (15), Wednesday(10), Thursday (20), Friday(15)


6. (a) 48 (b) Water park
(c) Sea side (d) 16


8. (a) 18000 (b) 14000
(c) 2007

In questions 1 to 5, out of the four options, only one is correct. write the correct answer.
1. Using tally marks, which one of the following represents the number eight:


2. The marks (out of 10) obtained by 28 students in a Mathematics test are listed as below:
8, 1, 2, 6, 5, 5, 5, 0, 1, 9, 7, 8, 0, 5, 8, 3, 0, 8, 10, 10, 3, 4, 8, 7, 8, 9,2, 0
The number of students who obtained marks more than or equal to 5 is
(A) 13 (B) 15 (C) 16 (D) 17

3. In question 2 above, the number of students who scored marks less than 4 is
(A) 15 (B) 13 (C) 12 (D) 10

4. The choices of the fruits of 42 students in a class are as follows:
A , O , B , M , A , G , B , G , A , G , B , M , A , G , M , A , B , G , M , B , A , O , M , O , G , B , O , M G , A , A , B , M , O , M , G , B , A , M , O , M , O,
where A, B, G, M and O stand for the fruits Apple, Banana, Grapes, Mango and Orange respectively.
Which two fruits are liked by an equal number of students?
(A) A and M (B) M and B (C) B and O (D) B and G

5. According to data of question 4, which fruit is liked by most of the students?
(A) O (B) G (C) M (D) A

In questions 6 to 13, state whether the given statements are true (T) or false (F).

6. In a bar graph, the width of bars may be unequal.

7. In a bar graph, bars of uniform width are drawn vertically only.

8. In a bar graph, the gap between two consecutive bars may not be the same.

9. In a bar graph, each bar (rectangle) represents only one value of the numerical data.

10. To represent the population of different towns using bar graph, it is convenient to take one unit length to represent one person.

11. Pictographs and bar graphs are pictorial representations of the numerical data.

12. An observation occurring five times in a data is recorded as | | | | |, using tally marks.

13. In a pictograph, if a symbol represents 50 books in a library shelf, then the symbol represents 25 books.

In questions 14 to 20, fill in the blanks to make the statements true:

14. A _______ is a collection of numbers gathered to give some meaningful information.

15. The data can be arranged in a tabular form using _______ marks.

16. A _______ represents data through pictures of objects.

17. In a bar graph, ________ can be drawn horizontally or vertically.

18. In a bar graph, bars of ________ width can be drawn horizontally or vertically with ________ spacing between them.

19. An observation occurring seven times in a data is represented as ________ using tally marks.

20. In a pictograph, if a symbol    represents 20 flowers in a basket then stands for ________ flowers.

21. On the scale of 1 unit length = 10 crore, the bar of length 6 units will represent ________ crore and of ——— units will represent 75 crore.

22. In an examination, the grades achieved by 30 students of a class are given below. Arrange these grades in a table using tally marks:
B, C, C, E, A, C, B, B, D, D, D, D, B, C, C, C, A, C, B, E, A, D, C, B, E, C, B, E, C, D

23. The number of two wheelers owned individually by each of 50 families are listed below. Make a table using tally marks.
1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 2, 1, 1, 4, 1, 3, 1, 1, 2, 1, 1, 1, 1,2, 3, 2, 1, 1
Find the number of families having two or more, two wheelers.

24. The lengths in centimetres (to the nearest centimetre) of 30 carrots are given as follows:
15, 22, 21, 20, 22,15, 15, 20, 20,15, 20, 18,20, 22, 21, 
20, 21, 18, 21, 18, 20, 18, 21, 18, 22, 20, 15, 21, 18, 20
Arrange the data given above in a table using tally marks and answer the following questions.
(a) What is the number of carrots which have length more than 20 cm?
(b) Which length of the carrots occur maximum number of times? Minimum number of times?

25. Thirty students were interviewed to find out what they want to be in future. Their responses are llisted as below: 
doctor, engineer, doctor, pilot, officer, doctor, engineer, doctor, pilot, officer, pilot, engineer, officer, pilot, doctor, engineer, pilot, officer, doctor, officer, doctor, pilot, engineer, doctor, pilot, officer, doctor,
pilot, doctor, engineer
Arrange the data in a table using tally marks.

26. Following are the choices of games of 40 students of Class VI: 
football, cricket, football, kho-kho, hockey, cricket, hockey, kho-kho, tennis, tennis, cricket, football, football, hockey, kho-kho, football, cricket, tennis, football, hockey, kho-kho, football, cricket, cricket,
football, hockey, kho-kho, tennis, football, hockey, cricket, football, hockey, cricket, football, kho- kho, football, cricket, hockey, football. 
(a) Arrange the choices of games in a table using tally marks.
(b) Which game is liked by most of the students?
(c) Which game is liked by minimum number of students?

27. Fill in the blanks in the following table which represents shirt size of 40 students of a school.

28. Following pictograph represents some surnames of people listed in the telephone directory of a city

Observe the pictograph and answer the following questions:
(a) How many people have surname ‘Roy’?
(b) Which surname appears the maximum number of times in the telephone directory?
(c) Which surname appears the least number of times in the directory?
(d) Which two surnames appear an equal number of times?

 

29. Students of Class VI in a school were given a task to count the number of articles made of different materials in the school. The information collected by them is represented as follows:

Observe the pictograph and answer the following questions:
(a) Which material is used in maximum number of articles?
(b) Which material is used in minimum number of articles?
(c) Which material is used in exactly half the number of articles as those made up of metal?
(d) What is the total number of articles counted by the students?

30. The number of scouts in a school is depicted by the following pictograph:

Observe the pictograph and answer the following questions:
(a) Which class has the minimum number of scouts?
(b) Which class has the maximum number of scouts?
(c) How many scouts are there in Class VI?
(d) Which class has exactly four times the scouts as that of Class X?
(e) What is the total number of scouts in the Classes VI to X?

 

31. A survey was carried out in a certain school to find out the popular school subjects among students of Classes VI to VIII. The data in this regard is displayed as pictograph given below:

(a) Which subject is most popular among the students?
(b) How many students like Mathematics?
(c) Find the number of students who like subjects other than Mathematics and Science.

32. The following pictograph depicts the information about the areas in sqkm (to nearest hundred) of some districts of Chhattisgarh State:

(a) What is the area of Koria district?
(b) Which two districts have the same area?
(c) How many districts have area more than 5000 square kilometres?

 

33. The number of bottles of cold drinks sold by a shopkeeper on six consecutive days is as follows:


Prepare a pictograph of the data using one symbol to represent 50 bottles.

34. The following table gives information about the circulation of newspapers (dailies) in a town in five languages.

Prepare a pictograph of the above data, using a symbol of your choice, each representing 1000 newspapers.

35. Annual expenditure of a company in the year 2007-2008 is given below:

Prepare a pictograph of the above data using an appropriate symbol to represent Rs 10 lakh.

36. The following bar graph shows the number of houses (out of 100) in a town using different types of fuels for cooking.

(a) Which fuel is used in maximum number of houses?
(b) How many houses are using coal as fuel?
(c) Suppose that the total number of houses in the town is 1 lakh. From the above graph estimate the number of houses using electricity.

37. The following bar graph represents the data for different sizes of shoes worn by the students in a school. Read the graph and answer the following questions.

(a) Find the number of students whose shoe sizes have been collected.
(b) What is the number of students wearing shoe size 6?
(c) What are the different  sizes of the shoes worn by the students?
(d) Which shoe size is worn by the maximum number of students?
(e) Which shoe size is worn by minimum number of students?
(f) State whether true or false:
The total number of students wearing shoe sizes 5 and 8 is the same as the number of students wearing shoe size 6.

ANSWER KEY

1. D   2. D    3. D   4. D   5. C