Index Number
Business Statistics B.Com Notes
CBCS Syllabus notes for all the universities
of India
Index Number and Its Features
Historically
index number was started in 1764 to compare the Italian price index in 1750
with price level in 1500. Originally index number was used to measure the
effect of changes in prices of various commodities but in today’s world it is
used in almost every field. Index number is also used to measure the pulse of
the economy as a whole and also called barometer of economic activity.
Index number is simply
an indicator of changes in prices and quantities. It is a specialized average
designed to measure the change in a group of related variables over a period of
time. It offers a device of estimating the relative changes of a variable when
measurement of actual changes is not possible.
It is also an indicator of inflationary or deflationary tendencies.
In
the words of Croxton and Cowden, “Index number is devices for measuring
differences in the magnitude of a group of related variables.”
Table of
Contents |
1.
Index Number Meaning and Its Features 2.
Uses and Limitations of Index Number 3.
Problems in Construction of Index Number 4.
Different Types of Index Number 5.
Various Methods for Calculation of Index Number 6.
Test of Adequacy of Index Number – Time & Factor reversal test 7.
Cost of living Index (CLI) 8.
Fixed Base and Chain Base Index Number 9.
Deflating, Splicing and Base Shifting 10.
Various Stock Market Indices including BSE Index and NSE Index |
Following are the various features of
index number:
1.
Measures of relative changes: Index number measure relative or percentage
changes in the variable over time.
2.
Quantitative expression: Index numbers offer a precise measurement of the
quantitative change in the concerned variable over time.
3. Specialised Average: Index number show
changes in terms of average.
4.
Index numbers measures the effect of changes over a period of time.
Uses and limitations of index number
Use of Index Number (why index number is called economic barometer)
Index
numbers are highly valuable in business and economics. They provide a good
basis for comparison as they are expressed in abstract units of measurement. Some
of the Use of Index number is listed below:
1.
Measurement
of change in the price level or the value of money: Index
number can be used to know the impact of the change in the value of money on
different sections of the society.
2.
Knowledge
of the change in standard of living: Index number helps to ascertain the
living standards of people. Money income may increase but if index number show
a decrease in the value if money. Living standard may even decline.
3.
Adjustment
in salaries and allowances: Cost of living index number is a useful guide
to the government and private enterprises to make necessary adjustment in
salaries and allowances of the workers.
4.
Information
regarding foreign trade: - Index of exports and imports provides
useful information regarding foreign trade.
5.
They are
important in forecasting future economic activity: Index
number is useful not only in studying the past and present workings of our
economy, but they are also important in forecasting future economic activity. Price
index numbers serve as a useful guide to the business community in planning and
decision making purposes.
Index
numbers are called barometers of economic activity because of the above
mentioned utilities.
Limitations of index number
Index number suffers from various limitations some of which are listed below:
1.
Not completely true: Index number
not fully true. The index number simply indicate arithmetical tendency of the
temporal changes in the variable.
2.
International comparison is not
possible: Different countries have different bass of index numbers; these
do not help international comparisons.
3.
Difference of time: With the passage
of time, it is difficult to make comparison of index number with the changing
time man’s habits.
4.
Limited use: Index numbers are
prepared with certain specific objective. If they are used for another purpose
they may lead to wrong conclusion.
5.
Lack of retail price index number:
Most of the index numbers are prepared on the basis of wholesaler prices. But
in real life, retail prices are most relevant, but it is difficult to collect
retail prices.
ALSO READ: CHAPTER WISE BUSINESS STATISTICS NOTES
Unit 1: Statistical Data and Descriptive Statistics
a. Nature and Classification of data
b. Measures of Central Tendency
d. Skewness, Moments and Kurtosis
Unit 2: Probability and Probability Distributions
b. Probability distributions:Binomial-Poisson-Normal
Unit 3: Simple Correlation and Regression Analysis
UNIT 6: Sampling Concepts, Sampling Distributions and Estimation
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Problems in construction of Index number
Before
constructing index numbers a careful thought must be given to the problems in
construction of index number which are listed below:
a.
Purpose of
index number: There are various types of index number, constructed with
different objectives. Before constructing an index number, one must define the
objective. Index number is not suitable for all purpose. Every index is of
limited and specific use. Failure to decide clearly the purpose of the index
would lead to confusion and wastage of time with no fruitful results.
b.
Selection of base year: Selection
of base year is another problem in the construction of index number. Base year
is the reference year with which comparisons of relative changes are made. It
may be a year, a month or a day. The indeed for base period is always taken as
100. The following points should be taken into consideration while selecting base year:
Ã˜
The base year should be a normal one.
Ã˜
The base period should not be too distant in
the past.
c.
Selection
of goods and services: Every goods and services cannot be included
while constructing an index number and hence one has to decide what commodities
to be included in constructing index number. The commodities should be selected
in such a manner that they are representative of the tastes, habits and
behaviour of the people for whom the index is meant.
d.
Selection
of price: After the commodities have been selected, the next problem is to
obtain price quotations for these commodities. Prices of many commodities vary
from place to place and even from shop to shop in the same market. Whether
wholesale or retail prices are used is also a problem in construction of index
number. Price to be selected must be made of representative places and persons.
e.
Selection
of appropriate weights: Weights are very important in constructing
index number. The term ‘weight’ refers to the relative importance of the different
items in the construction of the index. All items are not of equal importance
and hence it is necessary to devise some suitable method whereby the varying
importance of the different items is taken into account. This is done by
allocating weights. Thus we have broadly two types of indices- unweighted and
weighted indices.
f.
Choice of
average: Since index numbers are specialised averages a decision has to be
made as to which particular average i.e. simple, weighted or geometric average
should be used for constructing index numbers.
g.
Selection
of appropriate weights: The term weight refers to the relative
importance of the different items in the construction of the index. All items
are not of equal importance and hence it is necessary to devise some suitable
method whereby the varying importance of the different items is taken into
account.
h.
Selection
of appropriate formula: A large number of formulae have been devised
for constructing the index. The problem very often is that of selecting the
most appropriate formula.
Different types of index number in Statistics
Index
number is of three types: Price index, quantity index and value index.
Price Index: Price
index is a measure reflecting the average of the proportionate changes in the
prices of a specified set of goods and services between two periods of time.
Usually a price index is assigned a value of 100 in some selected base period
and the values of the index for other periods are intended to indicate the
average percentage change in prices compared with the base period.
Quantity Index: Quantity
index is a measure reflecting the average of the proportionate changes in the
quantities of a specified set of goods and services between two periods of
time. Usually a quantity index is assigned a value of 100 in some selected base
period and the values of the index for other periods are intended to indicate
the average percentage change in quantities compared with the base period. A
quantity index is built up from information on quantities such as the number or
total weight of goods or the number of services.
Value Index: Value
indeed is a measure reflecting the average of the proportionate changes in the
value of a specified set of goods and services between base year and current
year. Value of goods and services is obtained by multiplying prices and
quantities. Usually a value index is assigned a value of 100 in some selected
base period and the values of the index for other periods are intended to
indicate the average percentage change in values of compared with the base
period.
Methods for calculation of index number
Index
numbers are constructed in a number of ways. These are broadly divided under
two heads:
i)
Unweighted Indices ii)
Weighted indices
In
the unweighted indices weights are not expressly assigned whereas in the
weighted indices weights are assigned to the various items. Each of these types
may be further divided under two heads:
i) Unweighted indices:
a) Simple aggregative method and
b) Simple average of relatives.
a) Simple Aggregative method: Under this
method, Price index will be the aggregate of prices of the given period
expressed as a percentage of that of the base period. Here, I
= ∑P_{n }/_{ }∑P_{0 }x 100
There
are two main limitations of the simple aggregative index:
1.
In this type of index, the items with large unit prices will highly influence
the index number.
2.
No consideration is given to the relative importance of the commodities.
b) Simple Average of Price Relatives: A price
relative is ratio of current year price expressed as a percentage with respect
to the base year price. Under this method price relative of various items are
calculated individually and average of price relative are obtained by dividing
sum of price relative with number of items. Here, I = ∑ (P_{n }/_{ }P_{0 }x 100)/N
Merits
of this method:
1.
Extreme items do not influence the index as equal importance is given to all of
the items.
2.
The index is not influenced by the units in which prices are quoted.
Demerits
of this method:
1.
There is problem of selecting an appropriate average.
2.
This method gives equal importance to all the relatives which may not always be
correct.
ii) Weighted indices:
a) Weighted aggregative method
b) Weighted average of price relatives
a)
Weighted Aggregative method: These indices are of the simple aggregative type
with the fundamental difference that weights are assigned to the various items
included in the index. Some of the weighted aggregative index number is:
1.
Laspeyres Index: It is the most widely used index in practice. In this method
the base year quantities are taken as weights. The formula for constructing the
index is:
Laspeyres
Index number = ∑P_{n}Q_{0}/∑P_{0}Q_{0
}X 100
Merits of Laspeyres Index Method:
a.
It is easy and cheapest method to construct index number.
b.
The index number of each year can be compared directly because it considers
only base year quantities.
c.
Weights are only used for one year i.e., base year.
Demerits of Laspeyres Index Method:
a.
It does take into account changes in demand.
b.
It tends to overstate prices increases.
c.
It does not satisfy time and factor reversal test.
2.
Paasche’s Index: This method is not so popular alike Laspeyres index method. In
this method the current year quantities are taken as weights. The formula for
constructing the index is:
Paasche’s
Index number = ∑P_{n}Q_{n}/∑P_{0}Q_{n
}X 100
Merits of Paasche’s Index Method:
a.
It takes into account consumption patterns into account because it considers
current year’s quantity.
b.
It does not overstate prices increases.
Demerits of Paasche’s Index Method:
a.
This method is long and expensive as compared to Laspeyres method.
b.
This is not a pure index as prices and quantities changes.
c.
It does not satisfy time and factor reversal test.
3. Fishers ideal index number: Fishers
index is an ideal index number because it considers both current and base
year’s prices and quantities. It is the geometric mean of Laspeyres and Paasche’s
index and calculated as:
Fisher’s index is regarded as ideal
index because:
i)
It considers both base year and current year’s
price and quantity.
ii)
It satisfies both time reversal and factor
reversal test.
iii)
It is based on Geometric mean which is
theoretically considered to be the best average of constructing index number.
iv)
It is free from bias as it considers both
current year and base year price and qty.
vii) Prove that fisher’s index satisfies both time reversal test
and facto7r reversal test.
b) Weighted average of price relatives:
In
weighted average of prices relatives, AM or GM is used to find index number.
The following steps to be followed:
1.
Each items of the period for which the index number is calculated is converted
into percentage of the same item in the base period.
2.
Multiply the percentages as obtained in step 1 with the weights assigned to
each items.
3.
Add the results obtained in step 2.
4.
Divide the sum obtained in step 3 by the sum of the weights used. Symbolically:
I = ∑PV/∑V.
Test of adequacy of Index number formulae:
There are various formulas for
construction of index number and the problem is that the selection of most
appropriate formula for a given situation. In order to find the most
appropriate formula, the following tests are suggested:
a) Time reversal
b) Factor reversal test
c) Unit test
d) Circular test
a) Time Reversal Test: Time reversal
test is a test to determine whether a given period method will work both ways
in time, forward and backward. In the words of Fisher, “The test is that the
formula for calculating the index number should be such that it will give the
same ratio between one point of comparison and the other, no matter which of
the two is taken as base.” Only Fisher’s ideal index satisfied time reversal
test. Symbolically time reversal test can be written as: P_{01} * P_{10}
= 1
b) Factor Reversal Test: Factor reversal test holds that the product of a price index and the quantity index should be equal to the corresponding value index. In the words of Fisher, “Just as each formula should permit the interchange of the two items without giving inconsistent results so it ought to permit interchanging prices and quantities without giving inconsistent results, i.e. the two results multiply together should give the true value ratio.“ In other words the change in price multiplied by change in quantity should be equal the total change in value. Only Fisher’s ideal index satisfied time reversal test. Symbolically factor reversal test can be written as: P_{01} * Q_{01} =∑P_{1}Q_{1} / ∑P_{0}Q_{0}
c) Unit Test: Unit test requires
that the formula for construction an index number should be independent of the
units in which, or for which, prices and quantities are quoted. This formula is
satisfied by all the index number formulas except the simple aggregative index
method.
d) Circular Test: This
formula is similar to time reversal test method. This test is done where there
is a frequent shift in the base on which index number is calculated. If
comparison of more than two years is to be made, it is always desirable to
shift the original base to the previous year which enables us to adjust the
index values from period to period. A test of this shiftability of base is
called the circular test. Symbolically circular test can be written as: P_{01}*
P_{12}*P_{20}= 1
Cost of living index number (CLI) and its uses
Cost
of living index numbers generally represent the average change in prices over a
period of time, paid by a consumer for a fixed set of goods and services. It
measure the relative changes over time in the cost level require to maintain
similar standard of living. Items
contributing to consumer price index are generally:
i)
Food
ii)
Clothing
iii)
Fuel and Lighting
iv)
Housing
v)
Miscellaneous.
Uses
of cost of living index:-
i)
CLI numbers are used for adjustment of
dearness allowance to maintain the same standard of living.
ii)
It is used in fixing various economic
policies.
iii)
Its helps in measuring purchasing power of
money.
iv)
Real wages can be obtained with the help of
CLI numbers.
Steps in
Construction of a Consumer Price Index
The following are the steps in constructing a consumer price
index:
1) Decision is to be taken about the class of people for whom the
index is useful.
2) Next step is to conduct a family budget enquiry covering the
population group for whom the index is to be designed.
3) Obtaining price quotations from the localities in which the
class of people concerned reside.
4) After quotations have been collected at average price for each
of the items included in the index has to be worked out.
Methods of
Constructing the Index
After the above mentioned problems are decided, the following two
methods can be used to find cost of living index:
1) Aggregate expenditure method: When this method is applied the
quantities of commodities consumed by the particular group in the base year are
estimated which constitute the weights. The prices of commodities of various
groups for the current year are multiplied with the quantities consumer in the
base year. This method is similar to Laspeyres index number and formula for
calculation is given below:
Consumer
Price Index number = ∑P_{n}Q_{0}/∑P_{0}Q_{0
}X 100
2) Family
budget method: When this method is applied the family budgets of a large number
of people for whom the index is meant is carefully studied and the aggregate
expenditure of an average family on various items is estimated. These
constitute the weights. The weights are thus the value weights obtained by multiplying
the prices by quantities consumed. The price relatives of each commodity are
obtained and these price relatives are multiplied by the value weight for each
item and the product is divided by the sum of weights. Symbolically:
Consumer price index = ∑IW/∑W_{
}X 100
Fixed base method and Chain base method
Fixed
Base Method: Under this method index number is calculated with a fixed base
year. By this method the index number of a given year is not influenced by the
variation of prices of any other year.
Uses
and limitations of fixed base index number: Same as index number mentioned
above
Chain
Base Method: Under This method the price of any period or year is related with
the price of the immediate previous period or year. It has no relation with the
price of fixed based period.
Use of Chain base index number:-
i)
Direct comparison between two successive
periods is possible through link indices.
ii)
This method is useful when weights are
changing rapidly.
iii)
It facilitates introduction in new item
replacing old one.
iv)
It permits the introduction of new commodities
and the detection of old ones without necessitating either the recalculation of
entire series or other drastic changes.
Disadvantages
of Chain base index number:
i)
This Method involves immense calculation.
ii)
Easy interpretation is lacking.
Difference between
chain base method and fixed base method:
CHAIN BASE MEHTOD |
FIXED BASED MEHTOD |
No fixed base is there. |
Base Period is fixed. |
Immediately preceding period is taken as
base. |
Base period is arbitrarily chosen. |
Calculation is too long. |
Calculation is easy. |
During Calculation if there is any error
then the
Entire
calculation is wrong. |
This is not so in this method. |
If data for any period is missing then
subsequent chain indices cannot be computed. |
This problem does not arise here. |
Meaning of Base Shifting, Splicing and Deflating
Base Shifting
Due to change in business condition and economy, it becomes necessary to change the reference base period of an index number form one period to another without returning to the original row dat. This change of reference base period is called “Shifting the base”. The main reasons for base shifting are:
a) The previous base year is very old and it becomes useless for comparison.
b) It may be desired to compare several index number series which have been computed on different base periods particularly if the several series are to be shown on the same graph.
Splicing
Sometimes index number is available
for a longer period of time, and there is substantial revision in base period
which results into two indices. So there is a need to convert the two indices
into a continuous series. The procedure employed for this conversion is called
splicing. The below mentioned formula is used for splicing:
Spliced index no. = (Index No. of
current year X Old index of new base year)/100
Deflating
By deflating we mean making allowances
for the effect of changing price levels. A rise in price level means a
reduction in the purchasing power of money. For example, suppose the prices of
rice rises from Rs. 1,500 per quintal in 2010 to Rs. 3,000 per quintal in 2020,
it means that in 2020 one can buy only half of rice if he spends the same
amount which he was spending in 2010 or in other words, the value of rupee is
only 50 paise in 2020 as compared to 2010. Thus the value of a rupee is simply
the reciprocal of an appropriate price index written as a proportion. The
general expressions may be given thus:
Purchasing power of money = 1\Price
index
It shall be clear from above that
since the value of money goes down with rising prices the workers or the
salaried people are interested not so much in money wages as in real wages
i.e., not how much they earn but how much their income or wage will buy. For
calculating real wages, we can multiply money wages by a quantity measuring the
purchasing power of the rupee or we divide the cash wages by an appropriate
price index. This process is referred as deflating. The process of deflating
can be expressed in the form of formula as follows:
Real wages = (Money wages/Prices
index)*100
Real wage or income index no = Index
of money wages/Consumer price index
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