NCERT Textbook: Data Handling | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 44
3.1  REPRESENTATIVE VALUES
You might be aware of the term average and would have come across statements involving
the term ‘average’ in your day-to-day life:
l Isha spends on an average of about 5 hours daily for her studies.
l The average temperature at this time of the year is about 40 degree celsius.
l The average age of pupils in my class is 12 years.
l The average attendance of students in a school during its final examination was
98 per cent.
Many more of such statements could be there. Think about the statements given above.
Do you think that the child in the first statement studies exactly for 5 hours daily?
Or, is the temperature of the given place during that particular time always 40 degrees?
Or, is the age of each pupil in that class 12 years? Obviously not.
Then what do these statements tell you?
By average we understand that Isha, usually, studies for 5 hours. On some days, she
may study for less number of hours and on the other days she may study longer.
Similarly, the average temperature of 40 degree celsius, means that, very often, the
temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less
than 40 degree celsius and at other times, it may be more than 40°C.
Thus, we realise that average is a number that represents or shows the central tendency
of a group of observations or data. Since average lies between the highest and the lowest
value of the given data so, we say average is a measure of the central tendency of the
group of data. Different forms of data need different forms of representative or central
value to describe it. One of these representative values is the “Arithmetic mean”. Y ou
will learn about the other representative values in the later part of the chapter.
Data
Handling Chapter  3
2024-25
Page 2


MATHEMATICS 44
3.1  REPRESENTATIVE VALUES
You might be aware of the term average and would have come across statements involving
the term ‘average’ in your day-to-day life:
l Isha spends on an average of about 5 hours daily for her studies.
l The average temperature at this time of the year is about 40 degree celsius.
l The average age of pupils in my class is 12 years.
l The average attendance of students in a school during its final examination was
98 per cent.
Many more of such statements could be there. Think about the statements given above.
Do you think that the child in the first statement studies exactly for 5 hours daily?
Or, is the temperature of the given place during that particular time always 40 degrees?
Or, is the age of each pupil in that class 12 years? Obviously not.
Then what do these statements tell you?
By average we understand that Isha, usually, studies for 5 hours. On some days, she
may study for less number of hours and on the other days she may study longer.
Similarly, the average temperature of 40 degree celsius, means that, very often, the
temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less
than 40 degree celsius and at other times, it may be more than 40°C.
Thus, we realise that average is a number that represents or shows the central tendency
of a group of observations or data. Since average lies between the highest and the lowest
value of the given data so, we say average is a measure of the central tendency of the
group of data. Different forms of data need different forms of representative or central
value to describe it. One of these representative values is the “Arithmetic mean”. Y ou
will learn about the other representative values in the later part of the chapter.
Data
Handling Chapter  3
2024-25
DAT A HANDLING 45
3.2  ARITHMETIC MEAN
The most common representative value of a group of data is the arithmetic mean or the
mean. T o understand this in a better way, let us look at the following example:
T wo vessels contain 20 litres and 60 litres of milk respectively . What is the amount that
each vessel would have, if both share the milk equally? When we ask this question we are
seeking the arithmetic mean.
In the above case, the average or the arithmetic mean would be
Total quantity of milk
Number of vessels
 = 
20 60
2
+
 litres = 40 litres.
Thus, each vessels would have 40 litres of milk.
The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:
mean = 
Sum of all observations
number of observations
Consider these examples.
EXAMPLE 1 Ashish studies for 4 hours, 5 hours and 3 hours respectively on three
consecutive days. How many hours does he study daily on an average?
SOLUTION The average study time of Ashish would be
Total number of study hours
Number of days for which he stud died
 = 
4 5 3
3
+ +
 hours = 4 hours per day
Thus, we can say that Ashish studies for 4 hours daily on an average.
EXAMPLE 2 A batsman scored the following number of runs in six innings:
36, 35, 50, 46, 60, 55
Calculate the mean runs scored by him in an inning.
SOLUTION Total runs = 36 + 35 + 50 + 46 + 60 + 55 = 282.
T o find the mean, we find the sum of all the observations and divide it by the number of
observations.
Therefore, in this case, mean = 
282
6
 = 47. Thus, the mean runs scored in an inning are 47.
Where does the arithmetic mean lie
TRY THESE
How would you find the average of your study hours for the whole week?
2024-25
Page 3


MATHEMATICS 44
3.1  REPRESENTATIVE VALUES
You might be aware of the term average and would have come across statements involving
the term ‘average’ in your day-to-day life:
l Isha spends on an average of about 5 hours daily for her studies.
l The average temperature at this time of the year is about 40 degree celsius.
l The average age of pupils in my class is 12 years.
l The average attendance of students in a school during its final examination was
98 per cent.
Many more of such statements could be there. Think about the statements given above.
Do you think that the child in the first statement studies exactly for 5 hours daily?
Or, is the temperature of the given place during that particular time always 40 degrees?
Or, is the age of each pupil in that class 12 years? Obviously not.
Then what do these statements tell you?
By average we understand that Isha, usually, studies for 5 hours. On some days, she
may study for less number of hours and on the other days she may study longer.
Similarly, the average temperature of 40 degree celsius, means that, very often, the
temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less
than 40 degree celsius and at other times, it may be more than 40°C.
Thus, we realise that average is a number that represents or shows the central tendency
of a group of observations or data. Since average lies between the highest and the lowest
value of the given data so, we say average is a measure of the central tendency of the
group of data. Different forms of data need different forms of representative or central
value to describe it. One of these representative values is the “Arithmetic mean”. Y ou
will learn about the other representative values in the later part of the chapter.
Data
Handling Chapter  3
2024-25
DAT A HANDLING 45
3.2  ARITHMETIC MEAN
The most common representative value of a group of data is the arithmetic mean or the
mean. T o understand this in a better way, let us look at the following example:
T wo vessels contain 20 litres and 60 litres of milk respectively . What is the amount that
each vessel would have, if both share the milk equally? When we ask this question we are
seeking the arithmetic mean.
In the above case, the average or the arithmetic mean would be
Total quantity of milk
Number of vessels
 = 
20 60
2
+
 litres = 40 litres.
Thus, each vessels would have 40 litres of milk.
The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:
mean = 
Sum of all observations
number of observations
Consider these examples.
EXAMPLE 1 Ashish studies for 4 hours, 5 hours and 3 hours respectively on three
consecutive days. How many hours does he study daily on an average?
SOLUTION The average study time of Ashish would be
Total number of study hours
Number of days for which he stud died
 = 
4 5 3
3
+ +
 hours = 4 hours per day
Thus, we can say that Ashish studies for 4 hours daily on an average.
EXAMPLE 2 A batsman scored the following number of runs in six innings:
36, 35, 50, 46, 60, 55
Calculate the mean runs scored by him in an inning.
SOLUTION Total runs = 36 + 35 + 50 + 46 + 60 + 55 = 282.
T o find the mean, we find the sum of all the observations and divide it by the number of
observations.
Therefore, in this case, mean = 
282
6
 = 47. Thus, the mean runs scored in an inning are 47.
Where does the arithmetic mean lie
TRY THESE
How would you find the average of your study hours for the whole week?
2024-25
MATHEMATICS 46
THINK, DISCUSS AND WRITE
Consider the data in the above examples and think on the following:
l Is the mean bigger than each of the observations?
l Is it smaller than each observation?
Discuss with your friends. Frame one more example of this type
and answer the same questions.
Y ou will find that the mean lies inbetween the greatest and the smallest
observations.
In particular, the mean of two numbers will always lie between the two numbers. For
example the mean of 5 and 11 is 
5 11
2
8
+
=
, which lies between 5 and 11.
Can you use this idea to show that between any two fractional numbers, you can find
as many fractional numbers as you like. For example between 
1
2
 and 
1
4
 you have their
average 
1
2
1
4
2
+
 = 
3
8
 and then between 
1
2
 and 
3
8
, you have their average 
7
16
and so on.
1. Find the mean of your sleeping hours during one week.
2. Find atleast 5 numbers between 
1
2
 and 
1
3
.
3.2.1  Range
The difference between the highest and the lowest observation gives us an idea of the
spread of the observations. This can be found by subtracting the lowest observation from
the highest observation. We call the result the range of the observation. Look at the
following example:
EXAMPLE 3 The ages in years of 10 teachers of a school are:
  32, 41, 28, 54, 35, 26, 23, 33, 38, 40
(i) What is the age of the oldest teacher and that of the youngest teacher?
(ii) What is the range of the ages of the teachers?
(iii) What is the mean age of these teachers?
SOLUTION
(i) Arranging the ages in ascending order, we get:
23, 26, 28, 32, 33, 35, 38, 40, 41, 54
We find that the age of the oldest teacher is 54 years and the age of the youngest
teacher is 23 years.
TRY THESE
2024-25
Page 4


MATHEMATICS 44
3.1  REPRESENTATIVE VALUES
You might be aware of the term average and would have come across statements involving
the term ‘average’ in your day-to-day life:
l Isha spends on an average of about 5 hours daily for her studies.
l The average temperature at this time of the year is about 40 degree celsius.
l The average age of pupils in my class is 12 years.
l The average attendance of students in a school during its final examination was
98 per cent.
Many more of such statements could be there. Think about the statements given above.
Do you think that the child in the first statement studies exactly for 5 hours daily?
Or, is the temperature of the given place during that particular time always 40 degrees?
Or, is the age of each pupil in that class 12 years? Obviously not.
Then what do these statements tell you?
By average we understand that Isha, usually, studies for 5 hours. On some days, she
may study for less number of hours and on the other days she may study longer.
Similarly, the average temperature of 40 degree celsius, means that, very often, the
temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less
than 40 degree celsius and at other times, it may be more than 40°C.
Thus, we realise that average is a number that represents or shows the central tendency
of a group of observations or data. Since average lies between the highest and the lowest
value of the given data so, we say average is a measure of the central tendency of the
group of data. Different forms of data need different forms of representative or central
value to describe it. One of these representative values is the “Arithmetic mean”. Y ou
will learn about the other representative values in the later part of the chapter.
Data
Handling Chapter  3
2024-25
DAT A HANDLING 45
3.2  ARITHMETIC MEAN
The most common representative value of a group of data is the arithmetic mean or the
mean. T o understand this in a better way, let us look at the following example:
T wo vessels contain 20 litres and 60 litres of milk respectively . What is the amount that
each vessel would have, if both share the milk equally? When we ask this question we are
seeking the arithmetic mean.
In the above case, the average or the arithmetic mean would be
Total quantity of milk
Number of vessels
 = 
20 60
2
+
 litres = 40 litres.
Thus, each vessels would have 40 litres of milk.
The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:
mean = 
Sum of all observations
number of observations
Consider these examples.
EXAMPLE 1 Ashish studies for 4 hours, 5 hours and 3 hours respectively on three
consecutive days. How many hours does he study daily on an average?
SOLUTION The average study time of Ashish would be
Total number of study hours
Number of days for which he stud died
 = 
4 5 3
3
+ +
 hours = 4 hours per day
Thus, we can say that Ashish studies for 4 hours daily on an average.
EXAMPLE 2 A batsman scored the following number of runs in six innings:
36, 35, 50, 46, 60, 55
Calculate the mean runs scored by him in an inning.
SOLUTION Total runs = 36 + 35 + 50 + 46 + 60 + 55 = 282.
T o find the mean, we find the sum of all the observations and divide it by the number of
observations.
Therefore, in this case, mean = 
282
6
 = 47. Thus, the mean runs scored in an inning are 47.
Where does the arithmetic mean lie
TRY THESE
How would you find the average of your study hours for the whole week?
2024-25
MATHEMATICS 46
THINK, DISCUSS AND WRITE
Consider the data in the above examples and think on the following:
l Is the mean bigger than each of the observations?
l Is it smaller than each observation?
Discuss with your friends. Frame one more example of this type
and answer the same questions.
Y ou will find that the mean lies inbetween the greatest and the smallest
observations.
In particular, the mean of two numbers will always lie between the two numbers. For
example the mean of 5 and 11 is 
5 11
2
8
+
=
, which lies between 5 and 11.
Can you use this idea to show that between any two fractional numbers, you can find
as many fractional numbers as you like. For example between 
1
2
 and 
1
4
 you have their
average 
1
2
1
4
2
+
 = 
3
8
 and then between 
1
2
 and 
3
8
, you have their average 
7
16
and so on.
1. Find the mean of your sleeping hours during one week.
2. Find atleast 5 numbers between 
1
2
 and 
1
3
.
3.2.1  Range
The difference between the highest and the lowest observation gives us an idea of the
spread of the observations. This can be found by subtracting the lowest observation from
the highest observation. We call the result the range of the observation. Look at the
following example:
EXAMPLE 3 The ages in years of 10 teachers of a school are:
  32, 41, 28, 54, 35, 26, 23, 33, 38, 40
(i) What is the age of the oldest teacher and that of the youngest teacher?
(ii) What is the range of the ages of the teachers?
(iii) What is the mean age of these teachers?
SOLUTION
(i) Arranging the ages in ascending order, we get:
23, 26, 28, 32, 33, 35, 38, 40, 41, 54
We find that the age of the oldest teacher is 54 years and the age of the youngest
teacher is 23 years.
TRY THESE
2024-25
DAT A HANDLING 47
(ii) Range of the ages of the teachers = (54 – 23) years = 31 years
(iii) Mean age of the teachers
= 
23 26 28 32 33 35 38 40 41 54
10
+ + + + + + + + +
years
= 
350
10
years
 = 35 years
EXERCISE 3.1
1. Find the range of heights of any ten students of your class.
2. Organise the following marks in a class assessment, in a tabular form.
4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7
(i) Which number  is the highest? (ii) Which number is the lowest?
(iii) What is the range of the data? (iv) Find the arithmetic mean.
3. Find the mean of the first five whole numbers.
4. A cricketer scores the following runs in eight innings:
58, 76, 40, 35, 46, 45, 0, 100.
Find the mean score.
5. Following table shows the points of each player scored in four games:
Player Game Game Game Game
1 2 3 4
A 14 16 10 10
B 0 8 6 4
C 8 11 Did not 13
Play
Now answer the following questions:
(i) Find the mean to determine A ’s average number of points scored per game.
(ii) T o find the mean number of points per game for C, would you divide the total
points by 3 or by 4? Why?
(iii) B played in all the four games. How would you find the mean?
(iv) Who is the best performer?
6. The marks (out of 100) obtained by a group of students in a science test are 85, 76,
90, 85, 39, 48, 56, 95, 81 and 75. Find the:
(i) Highest and the lowest marks obtained by the students.
2024-25
Page 5


MATHEMATICS 44
3.1  REPRESENTATIVE VALUES
You might be aware of the term average and would have come across statements involving
the term ‘average’ in your day-to-day life:
l Isha spends on an average of about 5 hours daily for her studies.
l The average temperature at this time of the year is about 40 degree celsius.
l The average age of pupils in my class is 12 years.
l The average attendance of students in a school during its final examination was
98 per cent.
Many more of such statements could be there. Think about the statements given above.
Do you think that the child in the first statement studies exactly for 5 hours daily?
Or, is the temperature of the given place during that particular time always 40 degrees?
Or, is the age of each pupil in that class 12 years? Obviously not.
Then what do these statements tell you?
By average we understand that Isha, usually, studies for 5 hours. On some days, she
may study for less number of hours and on the other days she may study longer.
Similarly, the average temperature of 40 degree celsius, means that, very often, the
temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less
than 40 degree celsius and at other times, it may be more than 40°C.
Thus, we realise that average is a number that represents or shows the central tendency
of a group of observations or data. Since average lies between the highest and the lowest
value of the given data so, we say average is a measure of the central tendency of the
group of data. Different forms of data need different forms of representative or central
value to describe it. One of these representative values is the “Arithmetic mean”. Y ou
will learn about the other representative values in the later part of the chapter.
Data
Handling Chapter  3
2024-25
DAT A HANDLING 45
3.2  ARITHMETIC MEAN
The most common representative value of a group of data is the arithmetic mean or the
mean. T o understand this in a better way, let us look at the following example:
T wo vessels contain 20 litres and 60 litres of milk respectively . What is the amount that
each vessel would have, if both share the milk equally? When we ask this question we are
seeking the arithmetic mean.
In the above case, the average or the arithmetic mean would be
Total quantity of milk
Number of vessels
 = 
20 60
2
+
 litres = 40 litres.
Thus, each vessels would have 40 litres of milk.
The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:
mean = 
Sum of all observations
number of observations
Consider these examples.
EXAMPLE 1 Ashish studies for 4 hours, 5 hours and 3 hours respectively on three
consecutive days. How many hours does he study daily on an average?
SOLUTION The average study time of Ashish would be
Total number of study hours
Number of days for which he stud died
 = 
4 5 3
3
+ +
 hours = 4 hours per day
Thus, we can say that Ashish studies for 4 hours daily on an average.
EXAMPLE 2 A batsman scored the following number of runs in six innings:
36, 35, 50, 46, 60, 55
Calculate the mean runs scored by him in an inning.
SOLUTION Total runs = 36 + 35 + 50 + 46 + 60 + 55 = 282.
T o find the mean, we find the sum of all the observations and divide it by the number of
observations.
Therefore, in this case, mean = 
282
6
 = 47. Thus, the mean runs scored in an inning are 47.
Where does the arithmetic mean lie
TRY THESE
How would you find the average of your study hours for the whole week?
2024-25
MATHEMATICS 46
THINK, DISCUSS AND WRITE
Consider the data in the above examples and think on the following:
l Is the mean bigger than each of the observations?
l Is it smaller than each observation?
Discuss with your friends. Frame one more example of this type
and answer the same questions.
Y ou will find that the mean lies inbetween the greatest and the smallest
observations.
In particular, the mean of two numbers will always lie between the two numbers. For
example the mean of 5 and 11 is 
5 11
2
8
+
=
, which lies between 5 and 11.
Can you use this idea to show that between any two fractional numbers, you can find
as many fractional numbers as you like. For example between 
1
2
 and 
1
4
 you have their
average 
1
2
1
4
2
+
 = 
3
8
 and then between 
1
2
 and 
3
8
, you have their average 
7
16
and so on.
1. Find the mean of your sleeping hours during one week.
2. Find atleast 5 numbers between 
1
2
 and 
1
3
.
3.2.1  Range
The difference between the highest and the lowest observation gives us an idea of the
spread of the observations. This can be found by subtracting the lowest observation from
the highest observation. We call the result the range of the observation. Look at the
following example:
EXAMPLE 3 The ages in years of 10 teachers of a school are:
  32, 41, 28, 54, 35, 26, 23, 33, 38, 40
(i) What is the age of the oldest teacher and that of the youngest teacher?
(ii) What is the range of the ages of the teachers?
(iii) What is the mean age of these teachers?
SOLUTION
(i) Arranging the ages in ascending order, we get:
23, 26, 28, 32, 33, 35, 38, 40, 41, 54
We find that the age of the oldest teacher is 54 years and the age of the youngest
teacher is 23 years.
TRY THESE
2024-25
DAT A HANDLING 47
(ii) Range of the ages of the teachers = (54 – 23) years = 31 years
(iii) Mean age of the teachers
= 
23 26 28 32 33 35 38 40 41 54
10
+ + + + + + + + +
years
= 
350
10
years
 = 35 years
EXERCISE 3.1
1. Find the range of heights of any ten students of your class.
2. Organise the following marks in a class assessment, in a tabular form.
4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7
(i) Which number  is the highest? (ii) Which number is the lowest?
(iii) What is the range of the data? (iv) Find the arithmetic mean.
3. Find the mean of the first five whole numbers.
4. A cricketer scores the following runs in eight innings:
58, 76, 40, 35, 46, 45, 0, 100.
Find the mean score.
5. Following table shows the points of each player scored in four games:
Player Game Game Game Game
1 2 3 4
A 14 16 10 10
B 0 8 6 4
C 8 11 Did not 13
Play
Now answer the following questions:
(i) Find the mean to determine A ’s average number of points scored per game.
(ii) T o find the mean number of points per game for C, would you divide the total
points by 3 or by 4? Why?
(iii) B played in all the four games. How would you find the mean?
(iv) Who is the best performer?
6. The marks (out of 100) obtained by a group of students in a science test are 85, 76,
90, 85, 39, 48, 56, 95, 81 and 75. Find the:
(i) Highest and the lowest marks obtained by the students.
2024-25
MATHEMATICS 48
(ii) Range of the marks obtained.
(iii) Mean marks obtained by the group.
7. The enrolment in a school during six consecutive years was as follows:
1555, 1670, 1750, 2013, 2540, 2820
Find the mean enrolment of the school for this period.
8. The rainfall (in mm) in a city on 7 days of a certain week was recorded as follows:
Day Mon T ue W ed Thurs Fri Sat Sun
Rainfall
0.0 12.2 2.1 0.0 20.5 5.5 1.0
(in mm)
(i) Find the range of the rainfall in the above data.
(ii) Find the mean rainfall for the week.
(iii) On how many days was the rainfall less than the mean rainfall.
9. The heights of 10 girls were measured in cm and the results are as follows:
135, 150, 139, 128, 151, 132, 146, 149, 143, 141.
(i) What is the height of the tallest girl? (ii) What is the height of the shortest girl?
(iii) What is the range of the data? (iv) What is the mean height of the girls?
(v) How many girls have heights more than the mean height.
3.3  MODE
As we have said Mean is not the only measure of central tendency or the only form of
representative value. For different requirements from a data, other measures of central
tendencies are used.
Look at the following example
T o find out the weekly demand for different sizes of shirt, a shopkeeper kept records of sales
of sizes 90 cm, 95 cm, 100 cm, 105 cm, 110 cm. Following is the record for a week:
Size (in inches) 90 cm 95 cm 100 cm 105 cm 110 cm T otal
Number of Shirts Sold 8 22 32 37 6 105
If he found the mean number of shirts sold, do you think that he would be able to
decide which shirt sizes to keep in stock?
Mean of total shirts sold = 
Total number of shirts sold
Number of different sizes of shi irts
= =
105
5
21
Should he obtain 21 shirts of each size? If he does so, will he be able to cater to the
needs of the customers?
2024-25