NCERT Textbook - Measures of Central Tendency Notes | Study Economics Class 11 - Commerce

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 Page 1


Measures of Central Tendency
1. INTRODUCTION
In the previous chapter, you have read
about the tabular and graphic
representation of the data. In this
chapter, you will study the measures
of central tendency which is a
numerical method to explain the data
in brief. You can see examples of
summarising a large set of data in
day-to-day life, like average marks
obtained by students of a class in a test,
average rainfall in an area, average
production in a factory, average income
of persons living in a locality or
working in a firm, etc.
Baiju is a farmer. He grows food
grains in his land in a village called
Balapur in Buxar district of Bihar. The
village consists of 50 small farmers.
Baiju has 1 acre of land. You are
interested in knowing the economic
condition of small farmers of Balapur.
You want to compare the economic
CHAPTER
Studying this chapter should
enable you to:
• understand the need for
summarising a set of data by one
single number;
• recognise and distinguish
between the different types of
averages;
• learn to compute different types
of averages;
• draw meaningful conclusions
from a set of data;
• develop an understanding of
which type of average would be
the most useful in a particular
situation.
2020-21
Page 2


Measures of Central Tendency
1. INTRODUCTION
In the previous chapter, you have read
about the tabular and graphic
representation of the data. In this
chapter, you will study the measures
of central tendency which is a
numerical method to explain the data
in brief. You can see examples of
summarising a large set of data in
day-to-day life, like average marks
obtained by students of a class in a test,
average rainfall in an area, average
production in a factory, average income
of persons living in a locality or
working in a firm, etc.
Baiju is a farmer. He grows food
grains in his land in a village called
Balapur in Buxar district of Bihar. The
village consists of 50 small farmers.
Baiju has 1 acre of land. You are
interested in knowing the economic
condition of small farmers of Balapur.
You want to compare the economic
CHAPTER
Studying this chapter should
enable you to:
• understand the need for
summarising a set of data by one
single number;
• recognise and distinguish
between the different types of
averages;
• learn to compute different types
of averages;
• draw meaningful conclusions
from a set of data;
• develop an understanding of
which type of average would be
the most useful in a particular
situation.
2020-21
MEASURES OF CENTRAL TENDENCY 59
condition of Baiju in Balapur village.
For this, you may have to evaluate the
size of his land holding, by comparing
with the size of land holdings of other
farmers of Balapur. You may like to see
if the land owned by Baiju is –
1. above average in ordinary sense (see
the Arithmetic Mean)
2. above the size of what half the
farmers own (see the Median)
3. above what most of the farmers own
(see the Mode)
In order to evaluate Baiju’s relative
economic condition, you will have to
summarise the whole set of data of land
holdings of the farmers of Balapur. This
can be done by the use of central
tendency, which summarises the data
in a single value in such a way that this
single value can represent the entire
data. The measuring of central tendency
is a way of summarising the data in the
form of a typical or representative value.
There are several statistical
measures of central tendency or
“averages”. The three most commonly
used averages are:
• Arithmetic Mean
• Median
• Mode
You should note that there are two
more types of averages i.e. Geometric
Mean and Harmonic Mean, which are
suitable in certain situations.
However, the present discussion will
be limited to the three types of
averages mentioned above.
2. ARITHMETIC MEAN
Suppose the monthly income (in Rs) of
six families is given as:
1600, 1500, 1400, 1525, 1625, 1630.
The mean family income is
obtained by adding up the incomes
and dividing by the number of
families.
= 
= Rs 1,547
It implies that on an average, a
family earns Rs 1,547.
Arithmetic mean is the most
commonly used measure of central
tendency. It is defined as the sum of
the values of all observations divided
by the number of observations and is
usually denoted by 
X
 . In general, if
there are N observations as X
1
, X
2
,
 
X
3
,
..., X
N
, then the Arithmetic Mean is given
by
X + X + X +...+ X
N 1 2 3
X=
N
The right hand side can be written
as 
1
X
N
N
i i
?
=
. Here, i is an index
which takes successive values 1, 2,
3,...N.
For convenience, this will be written in
simpler form without the index i. Thus
X
X =
N
?
, where, SX = sum of all
observations and N =  total number of
observations.
2020-21
Page 3


Measures of Central Tendency
1. INTRODUCTION
In the previous chapter, you have read
about the tabular and graphic
representation of the data. In this
chapter, you will study the measures
of central tendency which is a
numerical method to explain the data
in brief. You can see examples of
summarising a large set of data in
day-to-day life, like average marks
obtained by students of a class in a test,
average rainfall in an area, average
production in a factory, average income
of persons living in a locality or
working in a firm, etc.
Baiju is a farmer. He grows food
grains in his land in a village called
Balapur in Buxar district of Bihar. The
village consists of 50 small farmers.
Baiju has 1 acre of land. You are
interested in knowing the economic
condition of small farmers of Balapur.
You want to compare the economic
CHAPTER
Studying this chapter should
enable you to:
• understand the need for
summarising a set of data by one
single number;
• recognise and distinguish
between the different types of
averages;
• learn to compute different types
of averages;
• draw meaningful conclusions
from a set of data;
• develop an understanding of
which type of average would be
the most useful in a particular
situation.
2020-21
MEASURES OF CENTRAL TENDENCY 59
condition of Baiju in Balapur village.
For this, you may have to evaluate the
size of his land holding, by comparing
with the size of land holdings of other
farmers of Balapur. You may like to see
if the land owned by Baiju is –
1. above average in ordinary sense (see
the Arithmetic Mean)
2. above the size of what half the
farmers own (see the Median)
3. above what most of the farmers own
(see the Mode)
In order to evaluate Baiju’s relative
economic condition, you will have to
summarise the whole set of data of land
holdings of the farmers of Balapur. This
can be done by the use of central
tendency, which summarises the data
in a single value in such a way that this
single value can represent the entire
data. The measuring of central tendency
is a way of summarising the data in the
form of a typical or representative value.
There are several statistical
measures of central tendency or
“averages”. The three most commonly
used averages are:
• Arithmetic Mean
• Median
• Mode
You should note that there are two
more types of averages i.e. Geometric
Mean and Harmonic Mean, which are
suitable in certain situations.
However, the present discussion will
be limited to the three types of
averages mentioned above.
2. ARITHMETIC MEAN
Suppose the monthly income (in Rs) of
six families is given as:
1600, 1500, 1400, 1525, 1625, 1630.
The mean family income is
obtained by adding up the incomes
and dividing by the number of
families.
= 
= Rs 1,547
It implies that on an average, a
family earns Rs 1,547.
Arithmetic mean is the most
commonly used measure of central
tendency. It is defined as the sum of
the values of all observations divided
by the number of observations and is
usually denoted by 
X
 . In general, if
there are N observations as X
1
, X
2
,
 
X
3
,
..., X
N
, then the Arithmetic Mean is given
by
X + X + X +...+ X
N 1 2 3
X=
N
The right hand side can be written
as 
1
X
N
N
i i
?
=
. Here, i is an index
which takes successive values 1, 2,
3,...N.
For convenience, this will be written in
simpler form without the index i. Thus
X
X =
N
?
, where, SX = sum of all
observations and N =  total number of
observations.
2020-21
60 STATISTICS FOR ECONOMICS
How Arithmetic Mean is Calculated
The calculation of arithmetic mean can
be studied under two broad categories:
1. Arithmetic Mean for Ungrouped
Data.
2. Arithmetic Mean for Grouped Data.
Arithmetic Mean for Series of
Ungrouped Data
Direct Method
Arithmetic mean by direct method is
the sum of all observations in a series
divided by the total number of
observations.
Example 1
Calculate Arithmetic Mean from the
data showing marks of students in a
class in an economics test: 40, 50, 55,
78, 58.
X
X
N
=
S
   
=
+ + + +
=
40 50 55 78 58
5
56 2 .
The average mark of students in the
economics test is 56.2.
Assumed Mean Method
If the number of observations in the
data is more and/or figures are large,
it is difficult to compute arithmetic
mean by direct method. The
computation can be made easier by
using assumed mean method.
In order to save time in calculating
mean from a data set containing a large
number of observations as well as large
numerical figures, you can use
assumed mean method. Here you
assume a particular figure in the data
as the arithmetic mean on the basis of
logic/experience. Then you may take
deviations of the said assumed mean
from each of the observation. You can,
then, take the summation of these
deviations and divide it by the number
of observations in the data. The actual
arithmetic mean is estimated by taking
the sum of the assumed mean and the
ratio of sum of deviations to number of
observations. Symbolically,
(HEIGHT IN INCHES)
2020-21
Page 4


Measures of Central Tendency
1. INTRODUCTION
In the previous chapter, you have read
about the tabular and graphic
representation of the data. In this
chapter, you will study the measures
of central tendency which is a
numerical method to explain the data
in brief. You can see examples of
summarising a large set of data in
day-to-day life, like average marks
obtained by students of a class in a test,
average rainfall in an area, average
production in a factory, average income
of persons living in a locality or
working in a firm, etc.
Baiju is a farmer. He grows food
grains in his land in a village called
Balapur in Buxar district of Bihar. The
village consists of 50 small farmers.
Baiju has 1 acre of land. You are
interested in knowing the economic
condition of small farmers of Balapur.
You want to compare the economic
CHAPTER
Studying this chapter should
enable you to:
• understand the need for
summarising a set of data by one
single number;
• recognise and distinguish
between the different types of
averages;
• learn to compute different types
of averages;
• draw meaningful conclusions
from a set of data;
• develop an understanding of
which type of average would be
the most useful in a particular
situation.
2020-21
MEASURES OF CENTRAL TENDENCY 59
condition of Baiju in Balapur village.
For this, you may have to evaluate the
size of his land holding, by comparing
with the size of land holdings of other
farmers of Balapur. You may like to see
if the land owned by Baiju is –
1. above average in ordinary sense (see
the Arithmetic Mean)
2. above the size of what half the
farmers own (see the Median)
3. above what most of the farmers own
(see the Mode)
In order to evaluate Baiju’s relative
economic condition, you will have to
summarise the whole set of data of land
holdings of the farmers of Balapur. This
can be done by the use of central
tendency, which summarises the data
in a single value in such a way that this
single value can represent the entire
data. The measuring of central tendency
is a way of summarising the data in the
form of a typical or representative value.
There are several statistical
measures of central tendency or
“averages”. The three most commonly
used averages are:
• Arithmetic Mean
• Median
• Mode
You should note that there are two
more types of averages i.e. Geometric
Mean and Harmonic Mean, which are
suitable in certain situations.
However, the present discussion will
be limited to the three types of
averages mentioned above.
2. ARITHMETIC MEAN
Suppose the monthly income (in Rs) of
six families is given as:
1600, 1500, 1400, 1525, 1625, 1630.
The mean family income is
obtained by adding up the incomes
and dividing by the number of
families.
= 
= Rs 1,547
It implies that on an average, a
family earns Rs 1,547.
Arithmetic mean is the most
commonly used measure of central
tendency. It is defined as the sum of
the values of all observations divided
by the number of observations and is
usually denoted by 
X
 . In general, if
there are N observations as X
1
, X
2
,
 
X
3
,
..., X
N
, then the Arithmetic Mean is given
by
X + X + X +...+ X
N 1 2 3
X=
N
The right hand side can be written
as 
1
X
N
N
i i
?
=
. Here, i is an index
which takes successive values 1, 2,
3,...N.
For convenience, this will be written in
simpler form without the index i. Thus
X
X =
N
?
, where, SX = sum of all
observations and N =  total number of
observations.
2020-21
60 STATISTICS FOR ECONOMICS
How Arithmetic Mean is Calculated
The calculation of arithmetic mean can
be studied under two broad categories:
1. Arithmetic Mean for Ungrouped
Data.
2. Arithmetic Mean for Grouped Data.
Arithmetic Mean for Series of
Ungrouped Data
Direct Method
Arithmetic mean by direct method is
the sum of all observations in a series
divided by the total number of
observations.
Example 1
Calculate Arithmetic Mean from the
data showing marks of students in a
class in an economics test: 40, 50, 55,
78, 58.
X
X
N
=
S
   
=
+ + + +
=
40 50 55 78 58
5
56 2 .
The average mark of students in the
economics test is 56.2.
Assumed Mean Method
If the number of observations in the
data is more and/or figures are large,
it is difficult to compute arithmetic
mean by direct method. The
computation can be made easier by
using assumed mean method.
In order to save time in calculating
mean from a data set containing a large
number of observations as well as large
numerical figures, you can use
assumed mean method. Here you
assume a particular figure in the data
as the arithmetic mean on the basis of
logic/experience. Then you may take
deviations of the said assumed mean
from each of the observation. You can,
then, take the summation of these
deviations and divide it by the number
of observations in the data. The actual
arithmetic mean is estimated by taking
the sum of the assumed mean and the
ratio of sum of deviations to number of
observations. Symbolically,
(HEIGHT IN INCHES)
2020-21
MEASURES OF CENTRAL TENDENCY 61
Let, A = assumed mean
X = individual observations
N = total numbers of observa-
tions
d = deviation of assumed mean
from individual observation,
i.e. d = X – A
Then sum of all deviations is taken
as Sd=S (X-A)
Then find 
Then add A and 
Sd
N
 to get 
X
Therefore, 
You should remember that any
value, whether existing in the data or
not, can be taken as assumed mean.
However, in order to simplify the
calculation, centrally located value in
the data can be selected as assumed
mean.
Example 2
The following data shows the weekly
income of 10 families.
Family
A B C D E F G H
I J
Weekly Income (in Rs)
850 700 100 750 5000 80 420 2500
400 360
Compute mean family income.
TABLE 5.1
Computation of Arithmetic Mean by
Assumed Mean Method
Families Income d = X – 850 d'
(X) = (X – 850)/10
A 850 0 0
B 700 –150 –15
C 100 –750 –75
D 750 –100 –10
E 5000 +4150 +415
F 80 –770 –77
G 420 –430 –43
H 2500 +1650 +165
I 400 –450 –45
J 360 –490 –49
11160 +2660            +266
Arithmetic Mean using assumed mean
method
X A
d
N
Rs
= + = +
=
S
850 2 660 10
1116
( , )/
, .
Thus, the average weekly income
of a family by both methods is
Rs 1,116. You can check this by using
the direct method.
Step Deviation Method
The calculations can be further
simplified by dividing all the deviations
taken from assumed mean by the
common factor ‘c’. The objective is to
avoid large numerical figures, i.e., if
d = X – A is very large, then find d'.
This can be done as follows:
d' = 
d
c
X A
c
=
-
.
The formula is given below:
X A
d
N
c = +
'
×
S
where d' = (X – A)/c,   c = common
factor, N = number of observations, A=
Assumed mean.
Thus, you can calculate the
arithmetic mean in the example  2, by
the step deviation method,
X = 850 + (266/10) × 10 = Rs 1,116.
2020-21
Page 5


Measures of Central Tendency
1. INTRODUCTION
In the previous chapter, you have read
about the tabular and graphic
representation of the data. In this
chapter, you will study the measures
of central tendency which is a
numerical method to explain the data
in brief. You can see examples of
summarising a large set of data in
day-to-day life, like average marks
obtained by students of a class in a test,
average rainfall in an area, average
production in a factory, average income
of persons living in a locality or
working in a firm, etc.
Baiju is a farmer. He grows food
grains in his land in a village called
Balapur in Buxar district of Bihar. The
village consists of 50 small farmers.
Baiju has 1 acre of land. You are
interested in knowing the economic
condition of small farmers of Balapur.
You want to compare the economic
CHAPTER
Studying this chapter should
enable you to:
• understand the need for
summarising a set of data by one
single number;
• recognise and distinguish
between the different types of
averages;
• learn to compute different types
of averages;
• draw meaningful conclusions
from a set of data;
• develop an understanding of
which type of average would be
the most useful in a particular
situation.
2020-21
MEASURES OF CENTRAL TENDENCY 59
condition of Baiju in Balapur village.
For this, you may have to evaluate the
size of his land holding, by comparing
with the size of land holdings of other
farmers of Balapur. You may like to see
if the land owned by Baiju is –
1. above average in ordinary sense (see
the Arithmetic Mean)
2. above the size of what half the
farmers own (see the Median)
3. above what most of the farmers own
(see the Mode)
In order to evaluate Baiju’s relative
economic condition, you will have to
summarise the whole set of data of land
holdings of the farmers of Balapur. This
can be done by the use of central
tendency, which summarises the data
in a single value in such a way that this
single value can represent the entire
data. The measuring of central tendency
is a way of summarising the data in the
form of a typical or representative value.
There are several statistical
measures of central tendency or
“averages”. The three most commonly
used averages are:
• Arithmetic Mean
• Median
• Mode
You should note that there are two
more types of averages i.e. Geometric
Mean and Harmonic Mean, which are
suitable in certain situations.
However, the present discussion will
be limited to the three types of
averages mentioned above.
2. ARITHMETIC MEAN
Suppose the monthly income (in Rs) of
six families is given as:
1600, 1500, 1400, 1525, 1625, 1630.
The mean family income is
obtained by adding up the incomes
and dividing by the number of
families.
= 
= Rs 1,547
It implies that on an average, a
family earns Rs 1,547.
Arithmetic mean is the most
commonly used measure of central
tendency. It is defined as the sum of
the values of all observations divided
by the number of observations and is
usually denoted by 
X
 . In general, if
there are N observations as X
1
, X
2
,
 
X
3
,
..., X
N
, then the Arithmetic Mean is given
by
X + X + X +...+ X
N 1 2 3
X=
N
The right hand side can be written
as 
1
X
N
N
i i
?
=
. Here, i is an index
which takes successive values 1, 2,
3,...N.
For convenience, this will be written in
simpler form without the index i. Thus
X
X =
N
?
, where, SX = sum of all
observations and N =  total number of
observations.
2020-21
60 STATISTICS FOR ECONOMICS
How Arithmetic Mean is Calculated
The calculation of arithmetic mean can
be studied under two broad categories:
1. Arithmetic Mean for Ungrouped
Data.
2. Arithmetic Mean for Grouped Data.
Arithmetic Mean for Series of
Ungrouped Data
Direct Method
Arithmetic mean by direct method is
the sum of all observations in a series
divided by the total number of
observations.
Example 1
Calculate Arithmetic Mean from the
data showing marks of students in a
class in an economics test: 40, 50, 55,
78, 58.
X
X
N
=
S
   
=
+ + + +
=
40 50 55 78 58
5
56 2 .
The average mark of students in the
economics test is 56.2.
Assumed Mean Method
If the number of observations in the
data is more and/or figures are large,
it is difficult to compute arithmetic
mean by direct method. The
computation can be made easier by
using assumed mean method.
In order to save time in calculating
mean from a data set containing a large
number of observations as well as large
numerical figures, you can use
assumed mean method. Here you
assume a particular figure in the data
as the arithmetic mean on the basis of
logic/experience. Then you may take
deviations of the said assumed mean
from each of the observation. You can,
then, take the summation of these
deviations and divide it by the number
of observations in the data. The actual
arithmetic mean is estimated by taking
the sum of the assumed mean and the
ratio of sum of deviations to number of
observations. Symbolically,
(HEIGHT IN INCHES)
2020-21
MEASURES OF CENTRAL TENDENCY 61
Let, A = assumed mean
X = individual observations
N = total numbers of observa-
tions
d = deviation of assumed mean
from individual observation,
i.e. d = X – A
Then sum of all deviations is taken
as Sd=S (X-A)
Then find 
Then add A and 
Sd
N
 to get 
X
Therefore, 
You should remember that any
value, whether existing in the data or
not, can be taken as assumed mean.
However, in order to simplify the
calculation, centrally located value in
the data can be selected as assumed
mean.
Example 2
The following data shows the weekly
income of 10 families.
Family
A B C D E F G H
I J
Weekly Income (in Rs)
850 700 100 750 5000 80 420 2500
400 360
Compute mean family income.
TABLE 5.1
Computation of Arithmetic Mean by
Assumed Mean Method
Families Income d = X – 850 d'
(X) = (X – 850)/10
A 850 0 0
B 700 –150 –15
C 100 –750 –75
D 750 –100 –10
E 5000 +4150 +415
F 80 –770 –77
G 420 –430 –43
H 2500 +1650 +165
I 400 –450 –45
J 360 –490 –49
11160 +2660            +266
Arithmetic Mean using assumed mean
method
X A
d
N
Rs
= + = +
=
S
850 2 660 10
1116
( , )/
, .
Thus, the average weekly income
of a family by both methods is
Rs 1,116. You can check this by using
the direct method.
Step Deviation Method
The calculations can be further
simplified by dividing all the deviations
taken from assumed mean by the
common factor ‘c’. The objective is to
avoid large numerical figures, i.e., if
d = X – A is very large, then find d'.
This can be done as follows:
d' = 
d
c
X A
c
=
-
.
The formula is given below:
X A
d
N
c = +
'
×
S
where d' = (X – A)/c,   c = common
factor, N = number of observations, A=
Assumed mean.
Thus, you can calculate the
arithmetic mean in the example  2, by
the step deviation method,
X = 850 + (266/10) × 10 = Rs 1,116.
2020-21
62 STATISTICS FOR ECONOMICS
Calculation of arithmetic mean for
Grouped data
Discrete Series
Direct Method
In case of discrete series,
frequency against each observation is
multiplied by the value of the
observation. The values, so obtained,
are summed up and divided by the total
number of frequencies. Symbolically,
X
fX
f
=
S
S
Where, S fX = sum of the product
of variables and frequencies.
S f = sum of frequencies.
Example 3
Plots in a housing colony come in only
three sizes: 100 sq. metre, 200 sq.
meters and 300 sq. metre and the
number of plots are respectively 200
50 and 10.
TABLE 5.2
Computation of Arithmetic Mean by
Direct Method
Plot size in  No. of d' = X–200
Sq. metre X    Plots (f) f X 100 fd'
100 200 20000 –1 –200
200 50 10000 0 0
300 10 3000 +1 10
260 33000 0 –190
Arithmetic mean using direct method,
X=
X
N
33000
260
126.92 Sq.metre
?
= =
Therefore, the mean plot size in the
housing  colony is 126.92 Sq. metre.
Assumed Mean Method
As in case of individual series the
calculations can be simplified by using
assumed mean method, as described
earlier, with a simple modification.
Since frequency (f) of each item is
given here, we multiply each deviation
(d) by the frequency to get fd. Then we
get S fd. The next step is to get the total
of all frequencies i.e. S f. Then find out
S fd/S f. Finally, the arithmetic mean
is calculated by 
X A
fd
f
= +
S
S
 using
assumed mean method.
Step Deviation Method
In this case, the deviations are divided
by the common factor ‘c’ which
simplifies the calculation. Here we
estimate d' = 
d
c
X A
c
=
-
 in order to
reduce the size of numerical figures for
easier calculation. Then get fd' and S fd'.
The formula for arithmetic mean using
step deviation method is given as,
X A
fd
f
c = +
'
×
S
S
Activity
• Find the mean plot size for the
data given in example 3, by
using step deviation and
assumed mean methods.
2020-21
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