This notes very useful statistics student

1. STATISTICS-205
BY BHAVY PATEL

2. index number-1
 Topic name
1. Introduction
2. Uses of index numbers
3. Problems in constructing index number
4. Concept of price index
5. Concept of quantity index
6. Concept of value index
7. Uses of price index
8. Fixed base & Chain base index
9. Whole sale price index number
10. Index number economics

3. *introduction*
 Historically the first index number was constructed in 1764 to compare the Italian price
index in 1750 with the price level in 1500.
 Index numbers are the indicators which reflect changes over a specified period of time in
(1) price of different commodities (2) industrial production (3) sales (4)imports and exports
(5)cost of living etc..
 These indicators are of paramount importance to the management personnel or any
government organisation or industrial concern for the purpose of reviewing position and
planning action
 If necessary and in the formulation of executive decisions they reflect the pulse of an
economy and serve as indicators of inflationary or deflationary tendencies.
 Just as in physics and chemistry barometer measure atmospheric of pressure of gases so
in economics index number measure of the pressure of economic behaviour and rightly
terms as economics barometers or barometer of economic activity.
 Important indices like index numbers of wholesale price ,industrial production, agricultural
production etc..
*Definition*

4.  Index number are statistical devices designed to measure the relative change in the level of
phenomenon (variable or group of a variable) with respect to time, geographical location, or
other characteristics such income, profession.
 In other word These are the index numbers which express the value of variable at any
given date called the “given period” as a percentage of the value of that variable at some
standard date called the “base period”.
 The variable may be
1.The price of particular commodity (e.g. silver, iron, etc..)
Or The group of commodities (e.g. consumer goods, etc..)
2. The volume of trads exports and imports, agricultural or industrial production.
3. The national income of country or cost of living of persons belonging of particular income.
Notes
 According to “WHELDON’’ An“ index number is a device which shows by its variation the
changes in a magnitude which is not capable of accurate measurement in it self or of direct
valuation in practice”.
 “Edgewarth” gave the classical definition of index number as follows “index number shows
by its variations the change in a magnitude which is not susceptible either of accurate
measurement in itself or of direct valuation in practice”.
 In the word of “LAWRENCE J KAPLAN” an index number is statistical measure of
fluctuations in a variable arranged in the form of a series and using a base period for making
comparisons.

5. *Uses of index number*
 Index numbers have a great use in deflating index numbers have the capacity to adjust the
primary data at different costs therefore it becomes easy to transform a nominal wage to real
wage
 Index numbers come up with several tends and tendencies the help with making important
conclusions; conclusions in cyclical forces and irregular forces
 We also use index numbers in areas like economics index numbers help with framing the
simple and suitable policies by providing guidelines. These guidelines also help to make
decisions relating to research quotients and so on
 The use of index numbers helps in tacking the future enhancement of the economic
activities most people use it in time series analysis to know more about the cycle
developments trends and variations etc..
 Index numbers measure the changes happening in the standards of living over a given
period of time
 UNIVARIATE INDEX
o An index which is calculated from single variable is called univariate index
 COMPOSITE INDEX
o An index which is calculated from group variable is called composite index
*CHARACTERISTICS OF INDEX NUMBERS *

6. INDEX NUMBERS ARE SPECIALIZED AVERAGE
 As we know an average is a single figure representing a group of figures. How ever to
obtain an average the items must be comparable. For example the average weight of man,
woman and children of a certain locality has no meaning at all. Further more the unit of
measurement must be same for all the items. How ever this is not so with index numbers.
Index numbers also one type of averages which shows in a single figure the change in two or
more series of different items which can be expressed in different units. For example while
constructing a consumer price index number the various items which are use in construction
are divided into broad heads namely food, clothing, fuel, lighting, house rent, and
miscellaneous which are expressed in different units.
Index numbers measuresthe net change in a group of related variables
 Since index numbers are essentially averages, they describe in one single figure the
increase or decrease in a group of related variables under study. The group of variables may
be prices of set of commodities, the volume of production in different sectors etc..
Index numbers measure the effect of changes over period of time
 Index numbers are most widely used for measuring changes over a period of time. For
example we can compare the agricultural production, industrial production, imports, exports,
wages etc in two different periods.
*PROBLEMS IN CONSTRUCTING INDEX NUMBERS*

7.  Purpose of index numbers.
An index number which is properly designed for a purpose can be most useful and
powerful tool. Thus the first and the foremost problem are to determine the purpose of
index numbers. If we know the purpose of the index numbers we can settle some related
problems. For example if the purpose of index number is to measure the changes in the
production of steel, the problem of selection of items is automatically settled.
i. Selection of commodities
After defining the purpose of index numbers, select only those commodities which are
related to that index. For example if the purpose of an index is to measure the cost of living
of low income group we should select only those commodities or items which are consumed
by persons belonging to this group and due care should be taken not to include the goods
which are utilized by the middle income group or high income group i.e. the goods like
cosmetics, other luxury goods like scooters, cars, refrigerators, television sets etc.
ii. Selection of base period
The period with which the comparisons of relative changes in the level of phenomenon are
made is termed as base period. The index for this period is always taken as 100. The
following are the basic criteria for the choice of the base period.
i) The base period must be a normal period i.e. a period frees from all sorts of
abnormalities or random fluctuations such as labor strikes, wars, floods, earthquakes etc.
ii) The base period should not be too distant from the given period. Since index numbers
are essential tools in business planning and economic policies the base period should not

8. be too far from the current period. For example for deciding increase in dearness
allowance at present there is no advantage in taking 1950 or 1960 as the base, the
comparison should be with the preceding year after which the DA has not been
increased.
iii) Fixed base or chain base .While selecting the base a decision has to be made as to
whether the base shall remain fixing or not i.e. whether we have fixed base or chain
base. In the fixed base method the year to which the other years are compared is
constant. On the other hand, in chain base method the prices of a year are linked with
those of the preceding year. The chain base method gives a better picture than what is
obtained by the fixed base method.
 How a base is selected if a normal period is not available?
Ans: Some times it is difficult to distinguish a year which can be taken as a normal year and
hence the average of a few years may be regarded as the value corresponding to the base
year.
iii. Data for index numbers
The data, usually the set of prices and of quantities consumed of the selected
commodities for different periods, places etc. constitute the raw material for the
construction of index numbers. The data should be collected from reliable sources such as
standard trade journals, official publications etc. for example for the construction of retail

9. price index numbers, the price quotations for the commodities should be obtained from
super bazaars, departmental stores etc. and not from wholesale dealers.
iv.Selection of appropriate weights
A decision as to the choice of weights is an important aspect of the construction of
index numbers. The problem arises because all items included in the construction are not of
equal importance. So proper weights should be attached to them to take into account their
relative importance. Thus there are two type of indices.
i) Un weighted indices- in which no specific weights are attached
ii) Weighted indices- in which appropriate weights are assigned to various items.
v. Choice of average.
Since index numbers are specialized averages, a choice of average to be used in their
construction is of great importance. Usually the following averages are used.
i) A.M
ii) G.M
iii) Median
Among these averages G.M is the appropriate average to be used. But in practice G.M is not
used as often as A.M because of its computational difficulties.
vi. Choice of formula.
A large variety of formulae are available to construct an index number. The problem
very often is that of selecting the appropriate formula. The choice of the formula would
depend not only on the purpose of the index but also on the data available.

10. *CONCEPT OF PRICE INDEX NUMBER*
 Here comparison is mode in respect of price. The prices of different commodities at a
certain time is collected and compared with the prices of the base period. Taking the price
of the base period as 100.we have already mentioned that the period of reference is
known as base period and any other period is known as current period.
 The major sub-divisions of price index number.
o Whole-sale price index number
o Retail price index number
 In case wholesale price index, wholesale prices of different commodities are collected
nation wise, from different market , different quotations , this prices index gives an idea of
general price level.
 The retail price index , generally gives us the change in retail prices of several common
items in market , several consumable articles etc.. it is one kind of consumer price index.
Another type of retail price index is cost of living index number. Though which purchasing
capacity of different sections of people can be chalked out. The section of people mean
the people of different castes community even their income group, high, medium or low
etc…
*CONCEPT OF QUANTITY INDEX *

11.  Through this index we can output or input level of firm, agricultural or industrial
production, exports and imports, etc.. the stock level at any time period with respect to
certain period of time as base year can be detected ultimately the net development within
certain period of time may be found helps to study the economic development.
*CONCEPT OF VALUE INDEX *
 Through this index total value of certain period is compared with the total value of the base
period. This index has very limited application in sales of a company inventories etc.. it helps
the future planning regarding valuation of certain articles.
*Special types*
 There are several factors where index number can be used any character like yield,
growth, profit etc.. can be characterised through index number. Any type of relative
change with respect to base period can be established through index number as business
activity index productivity index, industrial output index etc..
*USES OF PRICE INDEX NUMBER*
 One of the common uses of index number is price index number. Traditionally this index
helps us to measure of purchasing power of money. The increase in general price index
and purchasing power of money is inversely proportional

12. 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑝𝑜𝑤𝑒𝑟 =
100
𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑝𝑟𝑖𝑐𝑒 𝑖𝑛𝑑𝑒𝑥
 As for example price index in 2000 is 170 base year 1995 means in the base year 1995 it
was 100 and in 2000 it was become 170 then the purchasing power of a rupee
=
100
170
= 0.59
 That means a rupee in 2000 can purchase goods of such quantity or services that would be
purchased by 0.59 paise in 1995
 If price is increased the amount of commodity consumed is decreased and also the service
which money wages could buy will be decreased so the real wage will be decreased in this
situation index number helps to determine the changes in real wages by construction the
real wage index
𝑅𝑒𝑎𝑙 𝑤𝑎𝑔𝑒 𝑖𝑛𝑑𝑒𝑥 =
𝑚𝑜𝑛𝑒𝑦 𝑤𝑎𝑔𝑒 𝑖𝑛𝑑𝑒𝑥
𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑟 𝑝𝑟𝑖𝑐𝑒 𝑖𝑛𝑑𝑒𝑥
∗ 100
 Index number help us to measure the rate of exchange between the two sets of articles
comparing the price index numbers of two sets of articles A and B both with respect to
same base period we can adjust a parity rate between two groups. Let the current index
numbers of sets A and B are 140 and 160 respectively Then parity ratio

13. =
140
160
∗ 100
= 87.5
So that prices of goods in A are lower by 12.5% than those B
 Price index also help us in formulating and deciding appropriate business and economic
policies a business organisation or any government can take different policies time to time
observing the rising trend or decline of price index number.
 Insurance companies may take decision from the price index number regarding the future
rates of interest on the policies endowment.
 Government and private sectors decide from the price index about dearness and
allowances of the employees.
Notations Used
 P stands for price of commodity
 Q stands for quantity of commodity
 V stands for value of commodity
 Base year is denoted by the suffix 0
 Current year is denoted by the suffix 1
 P0 the price of commodity in the base period
 P1 the price of commodity in the current period

14.  P01 price index number for the current year relative to base year
 P10 price index number for the base year relative to current year
 Similarly symbols are used q and also in Q and V