NCERT Textbook - Index Numbers | Economics Class 11 - Commerce PDF Download

 Page 1


Index Numbers
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: 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
2024-25
Page 2


Index Numbers
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: 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
2024-25
INDEX NUMBERS 91
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.
2024-25
Page 3


Index Numbers
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: 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
2024-25
INDEX NUMBERS 91
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.
2024-25
92 STATISTICS FOR ECONOMICS
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 7.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B 5 6 20
C 4 5 25
D 2 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
4 6 5 3
2 5 4 2
100 138 5 =
+ + +
+ + +
× = .
Here, price is said to have risen by
38.5 per cent.
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
P q
P q
01
1 0
0 0
100 = ×
S
S
An index number becomes a
weighted index when the relative
importance of items is taken care of.
2024-25
Page 4


Index Numbers
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: 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
2024-25
INDEX NUMBERS 91
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.
2024-25
92 STATISTICS FOR ECONOMICS
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 7.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B 5 6 20
C 4 5 25
D 2 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
4 6 5 3
2 5 4 2
100 138 5 =
+ + +
+ + +
× = .
Here, price is said to have risen by
38.5 per cent.
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
P q
P q
01
1 0
0 0
100 = ×
S
S
An index number becomes a
weighted index when the relative
importance of items is taken care of.
2024-25
INDEX NUMBERS 93
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 7.2
Base period  Current period
Commodity Price Quantity Price Quantity
P
0
q
0
p
1
q
1
A 2 10 4 5
B 5 12 6 10
C 4 20 5 15
D 2 15 3 10
P
P q
P q
01
1 0
0 0
100 = ×
S
S
=
× + × + × + ×
× + × + × + ×
×
4 10 6 12 5 20 3 15
2 10 5 12 4 20 2 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
P q
01
1 1
0 1
100 = ×
S
S
=
× + × + × + ×
× + × + × + ×
×
4 5 6 10 5 15 3 10
2 5 5 10 4 15 2 10
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
2024-25
Page 5


Index Numbers
Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should Studying this chapter should
enable you to: enable you to: enable you to: enable you to: 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
2024-25
INDEX NUMBERS 91
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.
2024-25
92 STATISTICS FOR ECONOMICS
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 7.1
Commodity Base Current Percentage
period period change
price (Rs) price (Rs)
A 2 4 100
B 5 6 20
C 4 5 25
D 2 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
4 6 5 3
2 5 4 2
100 138 5 =
+ + +
+ + +
× = .
Here, price is said to have risen by
38.5 per cent.
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
P q
P q
01
1 0
0 0
100 = ×
S
S
An index number becomes a
weighted index when the relative
importance of items is taken care of.
2024-25
INDEX NUMBERS 93
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 7.2
Base period  Current period
Commodity Price Quantity Price Quantity
P
0
q
0
p
1
q
1
A 2 10 4 5
B 5 12 6 10
C 4 20 5 15
D 2 15 3 10
P
P q
P q
01
1 0
0 0
100 = ×
S
S
=
× + × + × + ×
× + × + × + ×
×
4 10 6 12 5 20 3 15
2 10 5 12 4 20 2 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
P q
01
1 1
0 1
100 = ×
S
S
=
× + × + × + ×
× + × + × + ×
×
4 5 6 10 5 15 3 10
2 5 5 10 4 15 2 10
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
2024-25
94 STATISTICS FOR ECONOMICS
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. Paasche’s price index of
132.1 is interpreted as a price rise of
32.1 per cent. 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 exmple
P
01
1
4
4
2
6
5
5
4
3
2
100 149 = + + +
?
?
?
?
?
?
× =
Thus,  the prices of the commodities
have risen by 49 per cent.
The weighted index  of price relatives
is the weighted arithmetic mean of price
relatives defined as
P
W
P
P
W
i
i
i
i
n
i
i
n
01
1
0
1
1
100
=
× ?
?
?
?
?
?
?
? =
=
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 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 7.3
Commodity Weight Base Current Price
in % year price year relative
price (in Rs)
(in Rs.)
A 40 2 4 200
B 30 5 6 120
C 20 4 5 125
D 10 2 3 150
2024-25