BBA 27/09/2021

Abdurrahman Sabir
Abdurrahmansabir27@gmail.com
MPhil Schoolar

Index number
An index number is a statistical measure of average change in variable or a group of variables
with respect to time or space. i.e., the variable may be the enrolment of students in an institution,
the cost of education of college students, price of a particular commodity or a group of
commodities, exports and imports and wages of workers etc.
Index numbers are obtained by expressing the data for various period or places as percentage of
some specific period or place selected for the purposes of comparison and mechanically called
the base. Index numbers may be computed on weekly or monthly basis but generally they are
computed on annual basis.
Note: In short, an index number is a device that measures the changes occurring in data from
time to time or from place to place
Problems involved in index number construction
a) To understand the purpose which an index is to serve, the purpose of the index may be to
compare the scores of two students or to measure the changes in the general price level etc.
b) To decide what data should be included. The data to be included should relate to purpose for
which the index is to be used. This step also involves the collection of data on score, wages,
production or whatever is being compared.
c) To decide what period should be chosen as the base period, i.e., the period with which the
other period are to be compared.
d) In case of composite index numbers, another problem is to decide what method of averaging
should be used to arrive at a single index for each period. The method of averaging usually
includes the system of weighting but sometimes one faces the problem of assigning some
explicit weights to the various items of the data so that their relative importance is taken into
account.
It is customary to denote the base period by the subscript 0, e.g. ( ) will denote the price
(or quantity) of a commodity in the base period, while subscripts, 1, 2, 3…. n denote the other
time period a chronological order. The average price of the base period chosen is then set equal
to. 100. Index numbers (or price relatives) for other periods denoted by are
then computed as relative to the base period. Thus the price relative for the given year n, will be

Price Relative =

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Note: price relative expresses the price of a commodity in a given year as a fraction of the price
in the base year. It is multiplied by 100 to make it a percentage but is usually expressed without
the percent symbol. Price relative is independent of any units of measurement
Types of Index Number by Nature

There are three types of Index numbers by nature:

 Price Index Number
 Quantity Index Number
 Value Index Number

Price Index Number

Price index number is a measure of the changes in the prices of certain
commodities with respect to time or space.

Quantity Index Number

Quantity index number or volume index number measures the changes in the
quantity or volume produced, consumed or sold of certain commodities with respect to time or
space.

Value Index Number

Value index number measures the changes in the value of commodities in given
period with reference to base period.

Where denotes the price of the given year

denotes the price of the base year
The Base Period (Year)
The period with which we like to compare the relative changes is known as reference
period or base period”. The base period may be a year, month, week or a day.
A base period (year) should be a normal year. By normal year we mean a year of
economic stability and free from crisis caused by wars, strikes, earthquake, floods etc. If a single
year of normal conditions is not available then the average of several years is used as the base
period.
There are two methods of selecting a base period:
 Fixed base method
 Chain base method

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Fixed Base Method

According to this method, a particular year is generally chosen as the base period which
remains unchanged during the life time of the index.
To compute index numbers by fixed base method, the value of the base year is taken as
100. Index numbers for other periods are computed by dividing the price of a given year by the
base year price and the results are multiplied by 100. Values so obtained are called price relatives
i.e.

Price Relative =

Chain base method

A chain base method is one in which the base period is not fixed but move with the given
year. That is, the relative are computed with the immediately preceding year as a base year much
relative are called link relatives, thus a

Link Relative

The link relatives are computed back to a fixed base by multiplying together all the link relative

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Types of Index Number by Treatment

i) Simple index number
Simple index numbers are those which is computed from single variable. A
simple index number may be very easily computed. The value of the variable for each
period divided by the value in the base period and the result is multiplied by 100.
Example: Index number of enrollment in colleges, index number of gold price etc.
For example: The wages paid to the workers in a certain institution in 1980 and 1983
were Rs.9.650 and Rs.1 1,580 respectively. Now taking 1980 as the base year and
1983 as the given year, we have

Wage index for 1983 =

= =

= 120
Results: The result indicates that if the wage 1evel in 1980 were denoted by100, it is
120 in 1983. In other words wages have increased by 20% for I983 by comparison
with 1980
Example: The price of wheat (per maund) is given for the year 1964 to 1973. Calculate simple
indexnumber using
1964 1965 1966 1967 1968 1969 1970 1971 1972 1973
Years
Prices 20 18 23 24 25 27 28 30 32 33
(Rs)
i) 1964 as base year
ii) Average of the prices for the first five years as base year
iii) Average of the prices of all the ten years as base year
Sol:
Simple index number
Years Prices(Rs) Average of the prices
1964 as base First five year as base For ten year as
year base year
1964 20 100 (20/22)*100=90.9 (20/26)*100=76.9
1965 18 (18/20)*100= 90 (18/22)*100=81.8 (18/26)*100=69.2
1966 23 (23/20)*100=115 104.5455 88.46154
1967 24 120 109.0909 92.30769
1968 25 125 113.6364 96.15385
1969 27 135 122.7273 103.8462

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1970 28 140 127.2727 107.6923

1971 30 150 136.3636 115.3846
1972 32 160 145.4545 123.0769
1973 33 165 150 126.9231

Example: The price of wheat (per maund) is given for the year 1960 to 1967. Calculate index
numbers by chain base method using 1960 as base year
Year 1960 1961 1962 1963 1964 1965 1966 1967
Prices (Rs) 40 45 48 50 52 54 56 60
Sol:
Year Prices (Rs) Link Relatives Chain Indices
1960 40 100 100
1961 45 (45/40)*100=112.5 (100*112.5)/100=112.5
1962 48 (48/45)*100=106.7 (112.5*106.7)/100=120.04
1963 50 104.2 125.08
1964 52 104 130.08
1965 54 103.8 135.02
1966 56 103.7 140.02
1967 60 107.1 149.96

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ii) Composite Index numbers

An index that is computed from two or more variables is referred to as a
composite index
Example: Composite index numbers are the wholesale price index numbers, consumer
price index number

Classification of composite index number

a) Un-weighted index number
b) Weight index number

a) Un-weighted index number

An index number that measures the changes in the prices of a
group of commodities when the relative importance i.e. weight of the commodities
are not taken in to account is called un- weighted index number or un-weighted
indices.

Un-weighted index numbers may also be classified as:

i) Simple Weighted Aggregative Index Numbers

Simple Aggregative Index is one that indicates the percentage change in
the aggregate prices of a number of commodities, (say k) at different periods. It is
obtained by dividing the sum of the base year prices of all commodities by the
sum of the base year prices of the same commodities and expressing the result as
a percentage. Symbolically, we have

Where; denotes the price index for the given year relative to the base year 0

denote the sum of price of the given year, and

; denote the sum of price of the base year

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ii) Simple Average of relatives

A simple average of price relatives is an index obtained by taking
the average of the price relatives of the given commodities for each year
and expressing the result as a percentage. If we take the arithmetic mean,
then we have = ( )

where k denote the number of commodities whose price relatives are thus combined
Example: The following table shows wholesale prices of wheat, rice and mutton
for the years 1972, 1973 and 1974. Compute the simple aggregative priceindices
for 1973 and 1974 using 1972 as a base.
Years Commodity (Prices in Rs.)
Wheat Rice Mutton
1972 30 80 240
1973 32 100 300
1974 37 110 400
Sol:
Years Commodity (Prices in Rs.) Total
Wheat Rice Mutton
1972 P0 30 80 240 350
1973 P1 32 100 300 432
1974 P2 37 110 400 547
So;
For 1973: = => => 123.43%

For 1974: = => => 156.29%

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Example: The following table shows wholesale prices of wheat, rice and mutton
for the years 1972, 1973 and 1974. Compute the simple aggregative priceindices
for 1973 and 1974 using 1972 as a base.
Years Commodity (Prices in Rs.)
Wheat Rice Mutton
1972 30 80 240
1973 32 100 300
1974 37 110 400
Sol:
Years Commodity (Prices in Rs.) Total Link relative Chain indices
Wheat Rice Mutton
1972 P0 30 80 240 350 100 100
1973 P1 32 100 300 432 (432/350)*10 (123.43*100/10
0= 123.43 0) = 123.23
1974 P2 37 110 400 547 (547/432)*10 (126.62*123.23
0=126.62 /100)= 156.29

b) Weighted index number

An index number that measures the changes in the prices of a group of
commodities when the relative importance i.e. weight of the commodities are
taken into account is called weighted index number or weighted indices.

Weighted index number may also be classified as:

i) Weighted Aggregative Index Numbers

There are various kinds of weighted aggregative index number;some of
them are discussed below:
Laspeyres Index Numbers
Since the Laspeyre’s formula use the base year prices (quantities) as
weight therefore it is called as base year weighted index. It is to be noted that, the
Laspeyre’s index is subject to upward bias (expected to overestimate)

Formula is:

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Paasche Index Numbers

Since the Paasche’s formula use the current year prices (quantities) as
weight therefore it is called the current year weighted index. It is to be noted that,
the Paasche’s index subject to downward bias (expected to underestimate).

Formula is:

Fisher (Ideal) Index Number

Fisher’s index number is the geometric mean of the Laspeyre’s and Paasche’s
Index Numbers i.e., √

Marshall-Edgeworth Index Number

( )
Formula is:
( )
This formula can also be written as:

Example: Construct the following weighted aggregative price index numbers for 1960 using
1956 as a base, from the given data.
Prices (Rs. Per md) Quantities (tons)
Commodity 1956 po 1960 p1 1956 qo 1960 q1
A 64 75 270 276
B 40 45 124 118
C 18 21 130 121
D 58 68 185 267

Find out (i) Laspeyre’s index

(ii) Paasche’s index
(iii) Fisher’s “Ideal” index
(iv) Marshall-Edgeworth index
Sol:

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Prices (Rs. Per md) Quantities (tons) po qo po q1 p1 qo p 1q 1

Commodity 1956 1960 1956 1960
po p1 qo q1
A 64 75 270 276 17280 17664 20250 20700
B 40 45 124 118 4960 4720 5580 5310
C 18 21 130 121 2340 2178 2730 2541
D 58 68 185 267 10730 15486 12580 18156
… … … … … 35310 40048 41140 46707

(i) Laspeyre’s index

, => => 116.5

(ii) Paasche’s index

=> => 116.6

(iii) Fisher’s “Ideal” index

√ => √ =>√

√ => => 116.5

(iv) Marshall-Edgeworth index

=>

=> => => 116.5

Example: Construct the following weighted aggregative price index numbers for 1960, 1961
using 1956 as a base, from the given data.
Prices (Rs. Per md) Quantities (tons)
Commodity 1956 po 1960 p1 1961 p2 1956 qo 1960 q1 1961 q2
A 64 75 80 270 276 290
B 40 45 41 124 118 144
C 18 21 20 130 121 137
D 58 68 56 185 267 355

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Find out (i) Laspeyre’s index

(ii) Paasche’s index
(iii) Fisher’s “Ideal” index
(iv) Marshall-Edgeworth index
Sol:
po qo po q 1 po q2 p1 qo p 1q 1 P2 q 0 P2 q 2
17280 17664 18560 20250 20700 21600 23200
4960 4720 5760 5580 5310 5084 5904
2340 2178 2466 2730 2541 2600 2740
10730 15486 20590 12580 18156 10360 19880
35310 40048 47376 41140 46707 39644 51724

(i) Laspeyre’s index

For 1960: , => => 116.5

For 1961: => => 112.6

(ii) Paasche’s index

For 1960: => => 116.6

For 1961: : => => 109.2

(iii) Fisher’s “Ideal” index

For 1960: √ => √

=√ => √

= 116.5

For 1961: √ => √

=√ => √

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= 1.1067 => 110.67

(iv) Marshall-Edgeworth index

For 1960: =>

=> => => 116.5

For 1961: =>

=> => 1.1049 => 110.49

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