NCERT Textbook - Index Numbers Commerce Notes | EduRev

Commerce : NCERT Textbook - Index Numbers Commerce Notes | EduRev

Created by: Pj Commerce Academy
 Page 1


Index Numbers
Studying this chapter should
enable you to:
• understand the meaning of the
term index number;
? become familiar with the use of
some widely used index
numbers;
? calculate an index number;
? appreciate its limitations.
1. INTRODUCTION
You have learnt in the previous
chapters how summary measures can
be obtained from a mass of data. Now
you will learn how to obtain summary
measures of change in a group of
related variables.
Rabi goes to the market after a long
gap. He finds that the prices of most
commodities have changed. Some
items have become costlier, while
others have become cheaper. On his
return from the market, he tells his
father about the change in price of the
each and every item, he bought. It is
bewildering to both. The industrial
sector consists of many subsectors.
Each of them is changing. The output
of some subsectors are rising, while it
is falling in some subsectors. The
changes are not uniform. Description
of the individual rates of change will
be difficult to understand. Can a
single figure summarise these
changes?  Look at the following  cases:
Case 1
An industrial worker was earning a
salary of Rs 1,000 in 1982. Today, he
CHAPTER
Page 2


Index Numbers
Studying this chapter should
enable you to:
• understand the meaning of the
term index number;
? become familiar with the use of
some widely used index
numbers;
? calculate an index number;
? appreciate its limitations.
1. INTRODUCTION
You have learnt in the previous
chapters how summary measures can
be obtained from a mass of data. Now
you will learn how to obtain summary
measures of change in a group of
related variables.
Rabi goes to the market after a long
gap. He finds that the prices of most
commodities have changed. Some
items have become costlier, while
others have become cheaper. On his
return from the market, he tells his
father about the change in price of the
each and every item, he bought. It is
bewildering to both. The industrial
sector consists of many subsectors.
Each of them is changing. The output
of some subsectors are rising, while it
is falling in some subsectors. The
changes are not uniform. Description
of the individual rates of change will
be difficult to understand. Can a
single figure summarise these
changes?  Look at the following  cases:
Case 1
An industrial worker was earning a
salary of Rs 1,000 in 1982. Today, he
CHAPTER
108 STATISTICS FOR ECONOMICS
earns Rs 12,000. Can his standard of
living be said to have risen 12 times
during this period? By how much
should his salary be  raised so that
he is as well off as before?
Case 2
You must be reading about the sensex
in the newspapers. The sensex
crossing 8000 points is, indeed,
greeted with euphoria. When, sensex
dipped 600 points recently, it eroded
investors’ wealth by Rs 1,53,690
crores. What exactly is sensex?
Case 3
The government says inflation rate will
not accelerate due to the rise in the
price  of petroleum products. How
does one measure inflation?
These are a sample of questions
you confront in your daily life. A study
of the index number helps  in
analysing these questions.
2. WHAT IS AN INDEX NUMBER
An index number is a statistical device
for measuring changes in the
magnitude of a group of related
variables. It  represents the general
trend of diverging ratios, from which
it is calculated. It is a measure of the
average change in a group of related
variables over two different situations.
The comparison may be between like
categories such as persons, schools,
hospitals etc. An index number also
measures changes in the value of the
variables such as prices of specified
list of commodities, volume of
production in different sectors of an
industry, production of various
agricultural crops, cost of living etc.
Conventionally, index numbers are
expressed in terms of percentage. Of
the two periods, the period with which
the comparison is to be made, is
known as the base period. The value
in the base period is  given the index
number 100. If you want to know how
much  the price has changed in 2005
from  the level in 1990, then 1990
becomes the base. The index number
of any period is in proportion with it.
Thus an index number of 250
indicates that the value is two and half
times that of the base period.
Price index numbers measure and
permit comparison of the prices of
certain goods. Quantity index
numbers  measure the changes in the
physical volume of  production,
construction or employment. Though
price index numbers are more widely
used, a production index is also an
important indicator of the level of the
output in the economy.
Page 3


Index Numbers
Studying this chapter should
enable you to:
• understand the meaning of the
term index number;
? become familiar with the use of
some widely used index
numbers;
? calculate an index number;
? appreciate its limitations.
1. INTRODUCTION
You have learnt in the previous
chapters how summary measures can
be obtained from a mass of data. Now
you will learn how to obtain summary
measures of change in a group of
related variables.
Rabi goes to the market after a long
gap. He finds that the prices of most
commodities have changed. Some
items have become costlier, while
others have become cheaper. On his
return from the market, he tells his
father about the change in price of the
each and every item, he bought. It is
bewildering to both. The industrial
sector consists of many subsectors.
Each of them is changing. The output
of some subsectors are rising, while it
is falling in some subsectors. The
changes are not uniform. Description
of the individual rates of change will
be difficult to understand. Can a
single figure summarise these
changes?  Look at the following  cases:
Case 1
An industrial worker was earning a
salary of Rs 1,000 in 1982. Today, he
CHAPTER
108 STATISTICS FOR ECONOMICS
earns Rs 12,000. Can his standard of
living be said to have risen 12 times
during this period? By how much
should his salary be  raised so that
he is as well off as before?
Case 2
You must be reading about the sensex
in the newspapers. The sensex
crossing 8000 points is, indeed,
greeted with euphoria. When, sensex
dipped 600 points recently, it eroded
investors’ wealth by Rs 1,53,690
crores. What exactly is sensex?
Case 3
The government says inflation rate will
not accelerate due to the rise in the
price  of petroleum products. How
does one measure inflation?
These are a sample of questions
you confront in your daily life. A study
of the index number helps  in
analysing these questions.
2. WHAT IS AN INDEX NUMBER
An index number is a statistical device
for measuring changes in the
magnitude of a group of related
variables. It  represents the general
trend of diverging ratios, from which
it is calculated. It is a measure of the
average change in a group of related
variables over two different situations.
The comparison may be between like
categories such as persons, schools,
hospitals etc. An index number also
measures changes in the value of the
variables such as prices of specified
list of commodities, volume of
production in different sectors of an
industry, production of various
agricultural crops, cost of living etc.
Conventionally, index numbers are
expressed in terms of percentage. Of
the two periods, the period with which
the comparison is to be made, is
known as the base period. The value
in the base period is  given the index
number 100. If you want to know how
much  the price has changed in 2005
from  the level in 1990, then 1990
becomes the base. The index number
of any period is in proportion with it.
Thus an index number of 250
indicates that the value is two and half
times that of the base period.
Price index numbers measure and
permit comparison of the prices of
certain goods. Quantity index
numbers  measure the changes in the
physical volume of  production,
construction or employment. Though
price index numbers are more widely
used, a production index is also an
important indicator of the level of the
output in the economy.
INDEX NUMBERS 109
3. CONSTRUCTION OF AN INDEX NUMBER
In the following sections, the
principles of constructing an index
number will be illustrated through
price index numbers.
Let us look at the following example:
Example 1
Calculation of simple aggregative price
index
TABLE 8.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B5 6 20
C4 5 25
D2 3 50
As you observe in this example, the
percentage changes are different for
every commodity. If the percentage
changes were the same for all four
items, a single measure would have
been sufficient to describe the change.
However, the percentage changes
differ and reporting the percentage
change for every item will be
confusing. It happens  when the
number of commodities is large, which
is common in any real  market
situation. A price index represents
these changes by a single numerical
measure.
There are two methods of
constructing an index number. It can
be computed by the aggregative
method and by the method of
averaging relatives.
The Aggregative Method
The formula for a simple aggregative
price index is
P
P
P
01
1
0
100 =¥
S
S
Where  P
1 
and P
0
 indicate the price
of the commodity  in the current
period and  base period respectively.
Using the data from example 1, the
simple aggregative price index is
P
01
46 5 3
2 542
100 138 5 =
++ +
++ +
¥= .
Here, price is said to have risen by
38.5 percent.
Do you know that such an index
is of limited use? The reason is that
the units of measurement of prices of
various commodities are  not  the
same. It is unweighted, because the
relative importance of the items has
not been properly reflected. The items
are treated as having equal
importance or weight. But what
happens in reality? In reality the items
purchased differ in order of
importance.  Food items occupy a
large proportion of our expenditure.
In that case  an equal rise in the price
of an item with large weight and that
of an item with low weight will have
different implications for the overall
change in the price index.
The formula for a weighted
aggregative price index is
P
Pq
Pq
01
11
01
100 =¥
S
S
An index number becomes a
weighted index  when the relative
Page 4


Index Numbers
Studying this chapter should
enable you to:
• understand the meaning of the
term index number;
? become familiar with the use of
some widely used index
numbers;
? calculate an index number;
? appreciate its limitations.
1. INTRODUCTION
You have learnt in the previous
chapters how summary measures can
be obtained from a mass of data. Now
you will learn how to obtain summary
measures of change in a group of
related variables.
Rabi goes to the market after a long
gap. He finds that the prices of most
commodities have changed. Some
items have become costlier, while
others have become cheaper. On his
return from the market, he tells his
father about the change in price of the
each and every item, he bought. It is
bewildering to both. The industrial
sector consists of many subsectors.
Each of them is changing. The output
of some subsectors are rising, while it
is falling in some subsectors. The
changes are not uniform. Description
of the individual rates of change will
be difficult to understand. Can a
single figure summarise these
changes?  Look at the following  cases:
Case 1
An industrial worker was earning a
salary of Rs 1,000 in 1982. Today, he
CHAPTER
108 STATISTICS FOR ECONOMICS
earns Rs 12,000. Can his standard of
living be said to have risen 12 times
during this period? By how much
should his salary be  raised so that
he is as well off as before?
Case 2
You must be reading about the sensex
in the newspapers. The sensex
crossing 8000 points is, indeed,
greeted with euphoria. When, sensex
dipped 600 points recently, it eroded
investors’ wealth by Rs 1,53,690
crores. What exactly is sensex?
Case 3
The government says inflation rate will
not accelerate due to the rise in the
price  of petroleum products. How
does one measure inflation?
These are a sample of questions
you confront in your daily life. A study
of the index number helps  in
analysing these questions.
2. WHAT IS AN INDEX NUMBER
An index number is a statistical device
for measuring changes in the
magnitude of a group of related
variables. It  represents the general
trend of diverging ratios, from which
it is calculated. It is a measure of the
average change in a group of related
variables over two different situations.
The comparison may be between like
categories such as persons, schools,
hospitals etc. An index number also
measures changes in the value of the
variables such as prices of specified
list of commodities, volume of
production in different sectors of an
industry, production of various
agricultural crops, cost of living etc.
Conventionally, index numbers are
expressed in terms of percentage. Of
the two periods, the period with which
the comparison is to be made, is
known as the base period. The value
in the base period is  given the index
number 100. If you want to know how
much  the price has changed in 2005
from  the level in 1990, then 1990
becomes the base. The index number
of any period is in proportion with it.
Thus an index number of 250
indicates that the value is two and half
times that of the base period.
Price index numbers measure and
permit comparison of the prices of
certain goods. Quantity index
numbers  measure the changes in the
physical volume of  production,
construction or employment. Though
price index numbers are more widely
used, a production index is also an
important indicator of the level of the
output in the economy.
INDEX NUMBERS 109
3. CONSTRUCTION OF AN INDEX NUMBER
In the following sections, the
principles of constructing an index
number will be illustrated through
price index numbers.
Let us look at the following example:
Example 1
Calculation of simple aggregative price
index
TABLE 8.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B5 6 20
C4 5 25
D2 3 50
As you observe in this example, the
percentage changes are different for
every commodity. If the percentage
changes were the same for all four
items, a single measure would have
been sufficient to describe the change.
However, the percentage changes
differ and reporting the percentage
change for every item will be
confusing. It happens  when the
number of commodities is large, which
is common in any real  market
situation. A price index represents
these changes by a single numerical
measure.
There are two methods of
constructing an index number. It can
be computed by the aggregative
method and by the method of
averaging relatives.
The Aggregative Method
The formula for a simple aggregative
price index is
P
P
P
01
1
0
100 =¥
S
S
Where  P
1 
and P
0
 indicate the price
of the commodity  in the current
period and  base period respectively.
Using the data from example 1, the
simple aggregative price index is
P
01
46 5 3
2 542
100 138 5 =
++ +
++ +
¥= .
Here, price is said to have risen by
38.5 percent.
Do you know that such an index
is of limited use? The reason is that
the units of measurement of prices of
various commodities are  not  the
same. It is unweighted, because the
relative importance of the items has
not been properly reflected. The items
are treated as having equal
importance or weight. But what
happens in reality? In reality the items
purchased differ in order of
importance.  Food items occupy a
large proportion of our expenditure.
In that case  an equal rise in the price
of an item with large weight and that
of an item with low weight will have
different implications for the overall
change in the price index.
The formula for a weighted
aggregative price index is
P
Pq
Pq
01
11
01
100 =¥
S
S
An index number becomes a
weighted index  when the relative
110 STATISTICS FOR ECONOMICS
importance of items is taken care of.
Here weights are quantity weights. To
construct a weighted aggregative
index, a well specified basket of
commodities is taken and its worth
each year is calculated. It thus
measures the changing value of a fixed
aggregate of goods. Since the total
value changes with a fixed basket, the
change is due to price change.
Various methods of calculating a
weighted aggregative index use
different baskets with respect to time.
Example 2
Calculation of weighted aggregative
price index
TABLE 8.2
Base period  Current period
Commodity Price Quantity Price Quality
P
0
q
0
p
1
q
1
A21045
B512610
C420515
D215310
P
Pq
Pq
01
11
01
100 =¥
S
S
=
¥+ ¥+ ¥ + ¥
¥+ ¥+ ¥ + ¥
¥
4 106 12 5 203 15
2 10 5 12 4202 15
100
=¥ =
257
190
100 135 3 .
This method uses the base period
quantities as weights. A weighted
aggregative price index using base
period quantities as weights, is  also
known as Laspeyre’s price index. It
provides an explanation to the
question that if the expenditure on
base period basket of commodities
was Rs 100, how much should be the
expenditure in the current period on
the same basket of commodities? As
you can see here, the value of base
period quantities has risen by 35.3 per
cent due to price rise. Using base
period quantities as weights, the price
is said to have risen by 35.3 percent.
Since the current period quantities
differ from the base period quantities,
the index number using current period
weights gives a different value of the
index number.
P
Pq
Pq
01
11
01
100 =¥
S
S
=
¥ + ¥+ ¥+ ¥
¥ + ¥+ ¥+ ¥
¥
45 6 105 15 3 10
2 5 510 4 15 215
100
=¥ =
185
140
100 132 1 .
It uses the current period
quantities as weights.  A weighted
aggregative price index using current
period quantities as weights is known
as Paasche’s price index.  It helps in
answering the question that, if  the
Page 5


Index Numbers
Studying this chapter should
enable you to:
• understand the meaning of the
term index number;
? become familiar with the use of
some widely used index
numbers;
? calculate an index number;
? appreciate its limitations.
1. INTRODUCTION
You have learnt in the previous
chapters how summary measures can
be obtained from a mass of data. Now
you will learn how to obtain summary
measures of change in a group of
related variables.
Rabi goes to the market after a long
gap. He finds that the prices of most
commodities have changed. Some
items have become costlier, while
others have become cheaper. On his
return from the market, he tells his
father about the change in price of the
each and every item, he bought. It is
bewildering to both. The industrial
sector consists of many subsectors.
Each of them is changing. The output
of some subsectors are rising, while it
is falling in some subsectors. The
changes are not uniform. Description
of the individual rates of change will
be difficult to understand. Can a
single figure summarise these
changes?  Look at the following  cases:
Case 1
An industrial worker was earning a
salary of Rs 1,000 in 1982. Today, he
CHAPTER
108 STATISTICS FOR ECONOMICS
earns Rs 12,000. Can his standard of
living be said to have risen 12 times
during this period? By how much
should his salary be  raised so that
he is as well off as before?
Case 2
You must be reading about the sensex
in the newspapers. The sensex
crossing 8000 points is, indeed,
greeted with euphoria. When, sensex
dipped 600 points recently, it eroded
investors’ wealth by Rs 1,53,690
crores. What exactly is sensex?
Case 3
The government says inflation rate will
not accelerate due to the rise in the
price  of petroleum products. How
does one measure inflation?
These are a sample of questions
you confront in your daily life. A study
of the index number helps  in
analysing these questions.
2. WHAT IS AN INDEX NUMBER
An index number is a statistical device
for measuring changes in the
magnitude of a group of related
variables. It  represents the general
trend of diverging ratios, from which
it is calculated. It is a measure of the
average change in a group of related
variables over two different situations.
The comparison may be between like
categories such as persons, schools,
hospitals etc. An index number also
measures changes in the value of the
variables such as prices of specified
list of commodities, volume of
production in different sectors of an
industry, production of various
agricultural crops, cost of living etc.
Conventionally, index numbers are
expressed in terms of percentage. Of
the two periods, the period with which
the comparison is to be made, is
known as the base period. The value
in the base period is  given the index
number 100. If you want to know how
much  the price has changed in 2005
from  the level in 1990, then 1990
becomes the base. The index number
of any period is in proportion with it.
Thus an index number of 250
indicates that the value is two and half
times that of the base period.
Price index numbers measure and
permit comparison of the prices of
certain goods. Quantity index
numbers  measure the changes in the
physical volume of  production,
construction or employment. Though
price index numbers are more widely
used, a production index is also an
important indicator of the level of the
output in the economy.
INDEX NUMBERS 109
3. CONSTRUCTION OF AN INDEX NUMBER
In the following sections, the
principles of constructing an index
number will be illustrated through
price index numbers.
Let us look at the following example:
Example 1
Calculation of simple aggregative price
index
TABLE 8.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B5 6 20
C4 5 25
D2 3 50
As you observe in this example, the
percentage changes are different for
every commodity. If the percentage
changes were the same for all four
items, a single measure would have
been sufficient to describe the change.
However, the percentage changes
differ and reporting the percentage
change for every item will be
confusing. It happens  when the
number of commodities is large, which
is common in any real  market
situation. A price index represents
these changes by a single numerical
measure.
There are two methods of
constructing an index number. It can
be computed by the aggregative
method and by the method of
averaging relatives.
The Aggregative Method
The formula for a simple aggregative
price index is
P
P
P
01
1
0
100 =¥
S
S
Where  P
1 
and P
0
 indicate the price
of the commodity  in the current
period and  base period respectively.
Using the data from example 1, the
simple aggregative price index is
P
01
46 5 3
2 542
100 138 5 =
++ +
++ +
¥= .
Here, price is said to have risen by
38.5 percent.
Do you know that such an index
is of limited use? The reason is that
the units of measurement of prices of
various commodities are  not  the
same. It is unweighted, because the
relative importance of the items has
not been properly reflected. The items
are treated as having equal
importance or weight. But what
happens in reality? In reality the items
purchased differ in order of
importance.  Food items occupy a
large proportion of our expenditure.
In that case  an equal rise in the price
of an item with large weight and that
of an item with low weight will have
different implications for the overall
change in the price index.
The formula for a weighted
aggregative price index is
P
Pq
Pq
01
11
01
100 =¥
S
S
An index number becomes a
weighted index  when the relative
110 STATISTICS FOR ECONOMICS
importance of items is taken care of.
Here weights are quantity weights. To
construct a weighted aggregative
index, a well specified basket of
commodities is taken and its worth
each year is calculated. It thus
measures the changing value of a fixed
aggregate of goods. Since the total
value changes with a fixed basket, the
change is due to price change.
Various methods of calculating a
weighted aggregative index use
different baskets with respect to time.
Example 2
Calculation of weighted aggregative
price index
TABLE 8.2
Base period  Current period
Commodity Price Quantity Price Quality
P
0
q
0
p
1
q
1
A21045
B512610
C420515
D215310
P
Pq
Pq
01
11
01
100 =¥
S
S
=
¥+ ¥+ ¥ + ¥
¥+ ¥+ ¥ + ¥
¥
4 106 12 5 203 15
2 10 5 12 4202 15
100
=¥ =
257
190
100 135 3 .
This method uses the base period
quantities as weights. A weighted
aggregative price index using base
period quantities as weights, is  also
known as Laspeyre’s price index. It
provides an explanation to the
question that if the expenditure on
base period basket of commodities
was Rs 100, how much should be the
expenditure in the current period on
the same basket of commodities? As
you can see here, the value of base
period quantities has risen by 35.3 per
cent due to price rise. Using base
period quantities as weights, the price
is said to have risen by 35.3 percent.
Since the current period quantities
differ from the base period quantities,
the index number using current period
weights gives a different value of the
index number.
P
Pq
Pq
01
11
01
100 =¥
S
S
=
¥ + ¥+ ¥+ ¥
¥ + ¥+ ¥+ ¥
¥
45 6 105 15 3 10
2 5 510 4 15 215
100
=¥ =
185
140
100 132 1 .
It uses the current period
quantities as weights.  A weighted
aggregative price index using current
period quantities as weights is known
as Paasche’s price index.  It helps in
answering the question that, if  the
INDEX NUMBERS 111
the current period basket of
commodities  was consumed in the
base period and if we were spending
Rs 100 on it, how much should be the
expenditure in current period on the
same basket of commodities. A
Paasche’s price index of 132.1 is
interpreted as a price rise of 32.1
percent. Using current period weights,
the price is said to have risen by 32.1
per cent.
Method of Averaging relatives
When there is only one commodity, the
price index is the ratio of the price of
the commodity in the current period
to that in the base period, usually
expressed in percentage terms. The
method of averaging relatives takes
the average of these relatives when
there are many commodities. The
price index number using price
relatives is defined as
P
n
p
p
01
1
0
1
100 =¥ S
where  P
1
 and  P
o
 indicate the price of
the ith commodity  in the current
period and  base period respectively.
The  ratio  (P
1
/P
0
) × 100 is also referred
to as price relative of the commodity.
n stands for the number of
commodities. In the current
example
P
01
1
4
4
2
6
5
5
4
3
2
100 149 =+++
Ê
Ë
Á
ˆ
¯
˜
¥=
Thus  the prices of the commodities
have risen by 49 percent.
The weighted index  of price
relatives  is the weighted arithmetic
mean of price relatives defined as
 
P
W
P
P
W
01
1
0
100
=
¥
Ê
Ë
Á
ˆ
¯
˜
S
S
where W = Weight.
In a  weighted price relative index
weights may be determined by the
proportion or percentage of
expenditure on them in total
expenditure during the base period.
It can also refer to the current period
depending on the formula  used. These
are, essentially, the value shares of
different commodities in the total
expenditure. In general the base
period weight is preferred to the
current period weight. It is because
calculating the weight every year is
inconvenient. It also refers to the
changing values of different baskets.
They are strictly not comparable.
Example 3 shows the type of
information one needs for calculating
weighted price index.
Example 3
Calculation of weighted price relatives
index
TABLE 8.3
Commodity Base Current Price Weight
year year price relative in %
price (in Rs)
(in Rs.)
A 2 4 200 40
B 5 6 120 30
C 4 5 125 20
D 2 3 150 10
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