NCERT Textbook - Measures of Central Tendency Commerce Notes | EduRev

Statistics for Economics - Class XI

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Commerce : NCERT Textbook - Measures of Central Tendency Commerce Notes | EduRev

 Page 1


Measures of Central Tendency
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
most useful in a particular
situation.
1. INTRODUCTION
In the previous chapter, you have read
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
Page 2


Measures of Central Tendency
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
most useful in a particular
situation.
1. INTRODUCTION
In the previous chapter, you have read
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
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 below)
2 . above the size of what half the
farmers own (see the Median
below)
3 . above what most of the farmers
own (see the Mode below)
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 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
1600 1500 1400 1525 1625 1630
6
+++++
= 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
XX X X
N
X
N
N
=
+++ +
=
12 3
...
S
Where, 
SX
 = sum of all observa-
tions and N =  total number of obser-
vations.
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.
Page 3


Measures of Central Tendency
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
most useful in a particular
situation.
1. INTRODUCTION
In the previous chapter, you have read
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
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 below)
2 . above the size of what half the
farmers own (see the Median
below)
3 . above what most of the farmers
own (see the Mode below)
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 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
1600 1500 1400 1525 1625 1630
6
+++++
= 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
XX X X
N
X
N
N
=
+++ +
=
12 3
...
S
Where, 
SX
 = sum of all observa-
tions and N =  total number of obser-
vations.
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.
60 STATISTICS FOR ECONOMICS
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 marks of students in
the economics test are 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 of calculation
of 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 observa-
tions. Symbolically,
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
(HEIGHT IN INCHES)
Page 4


Measures of Central Tendency
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
most useful in a particular
situation.
1. INTRODUCTION
In the previous chapter, you have read
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
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 below)
2 . above the size of what half the
farmers own (see the Median
below)
3 . above what most of the farmers
own (see the Mode below)
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 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
1600 1500 1400 1525 1625 1630
6
+++++
= 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
XX X X
N
X
N
N
=
+++ +
=
12 3
...
S
Where, 
SX
 = sum of all observa-
tions and N =  total number of obser-
vations.
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.
60 STATISTICS FOR ECONOMICS
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 marks of students in
the economics test are 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 of calculation
of 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 observa-
tions. Symbolically,
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
(HEIGHT IN INCHES)
MEASURES OF CENTRAL TENDENCY 61
Then sum of all deviations is taken
as SS dX A =- ()
Then find 
Sd
N
Then add A and 
Sd
N
 to get 
X
Therefore, 
XA
d
N
=+
S
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
AB CD E F G H
IJ
Weekly Income (in Rs)
850 700 100 7505000 80 4202500
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
XA
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
c
XA
C
=
-
.
The formula is given below:
XA
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.
Calculation of arithmetic mean for
Grouped data
Discrete Series
Direct Method
In case of discrete series, frequency
against each of the observations is
Page 5


Measures of Central Tendency
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
most useful in a particular
situation.
1. INTRODUCTION
In the previous chapter, you have read
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
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 below)
2 . above the size of what half the
farmers own (see the Median
below)
3 . above what most of the farmers
own (see the Mode below)
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 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
1600 1500 1400 1525 1625 1630
6
+++++
= 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
XX X X
N
X
N
N
=
+++ +
=
12 3
...
S
Where, 
SX
 = sum of all observa-
tions and N =  total number of obser-
vations.
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.
60 STATISTICS FOR ECONOMICS
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 marks of students in
the economics test are 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 of calculation
of 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 observa-
tions. Symbolically,
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
(HEIGHT IN INCHES)
MEASURES OF CENTRAL TENDENCY 61
Then sum of all deviations is taken
as SS dX A =- ()
Then find 
Sd
N
Then add A and 
Sd
N
 to get 
X
Therefore, 
XA
d
N
=+
S
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
AB CD E F G H
IJ
Weekly Income (in Rs)
850 700 100 7505000 80 4202500
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
XA
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
c
XA
C
=
-
.
The formula is given below:
XA
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.
Calculation of arithmetic mean for
Grouped data
Discrete Series
Direct Method
In case of discrete series, frequency
against each of the observations is
62 STATISTICS FOR ECONOMICS
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 product of
variables and frequencies.
S f = sum of frequencies.
Example 3
Calculate mean farm size of
cultivating households in a village for
the following data.
Farm Size (in acres):
64 63 62 61 60 59
No. of Cultivating Households:
8 18 12 976
TABLE 5.2
 Computation of Arithmetic Mean by
Direct Method
Farm Size No. of X d f d
(X ) cultivating (1 × 2) (X - 62)(2 × 4)
in acres households(f)
( 1)( 2)( 3)( 4)( 5 )
64 8 512 +2 +16
63 18 1134 +1 +18
62 12 744 0 0
61 9 549 –1 –9
60 7 420 –2 –14
59 6 354 –3 –18
60 3713 –3 –7
Arithmetic mean using direct method,
X
fX
f
acres == =
S
S
3717
60
61 88 .
Therefore, the mean farm size in a
village is 61.88 acres.
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. Sf. Then
find out S fd/Sf. Finally the
arithmetic mean is calculated by
XA
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
XA
C
=
-
 in order to
reduce the size of numerical figures
for easier calculation. Then get fd' and
Sfd'. Finally the formula for step
deviation method is given as,
XA
fd
f
c =+
¢
·
S
S
Activity
• Find the mean farm size for the
data given in example 3, by using
step deviation and assumed
mean methods.
Continuous Series
Here, class intervals are given. The
process of calculating arithmetic mean
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