CHAPTER 18

INDEX NUMBERS

Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
 Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
 Different methods of computing index number.

CHAPTER OVERVIEW

Index Numbers

Issues Involved Types of Construction of Chain Index Usefulness of Tests of

Index Numbers Index Numbers Index Numbers Numbers Index Numbers Adequacy

Price Index Quantity Index Value Index Deflating Index Splicing of Index
Numbers Numbers Numbers Numbers Numbers

Unit Times Factor Circular

Test Reversal Text Reversal Text Test

© The Institute of Chartered Accountants of India

18.2 STATISTICS

Index numbers are convenient devices for measuring relative changes of differences from time
to time or from place to place. Just as the arithmetic mean is used to represent a set of values, an
index number is used to represent a set of values over two or more different periods or localities.
The basic device used in all methods of index number construction is to average the relative
change in either quantities or prices since relatives are comparable and can be added even though
the data from which they were derived cannot themselves be added. For example, if wheat
production has gone up to 110% of the previous year’s producton and cotton production has
gone up to 105%, it is possible to average the two percentages as they have gone up by 107.5%.
This assumes that both have equal weight; but if wheat production is twice as important as
cotton, percentage should be weighted 2 and 1. The average relatives obtained through this process
are called the index numbers.
Definition: An index number is a ratio of two or more time periods are involved, one of which is
the base time period. The value at the base time period serves as the standard point of comparison.
Example: NSE, BSE, WPI, CPI etc.
An index time series is a list of index numbers for two or more periods of time, where each index
number employs the same base year.
Relatives are derived because absolute numbers measured in some appropriate unit, are often of
little importance and meaningless in themselves. If the meaning of a relative figure remains
ambiguous, it is necessary to know the absolute as well as the relative number.
Our discussion of index numbers is confined to various types of index numbers, their uses, the
mathematical tests and the principles involved in the construction of index numbers.
Index numbers are studied here because some techniques for making forecasts or inferences
about the figures are applied in terms of index number. In regression analysis, either the
independent or dependent variable or both may be in the form of index numbers. They are less
unwieldy than large numbers and are readily understandable.
These are of two broad types: simple and composite. The simple index is computed for one
variable whereas the composite is calculated from two or more variables. Most index numbers
are composite in nature.

Following are some of the important criteria/problems which have to be faced in the construction
of index Numbers.
Selection of data: It is important to understand the purpose for which the index is used. If it is used
for purposes of knowing the cost of living, there is no need of including the prices of capital goods
which do not directly influence the living.
Index numbers are often constructed from the sample. It is necessary to ensure that it is
representative. Random sampling, and if need be, a stratified random sampling can ensure this.

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 INDEX NUMBERS 18.3

It is also necessary to ensure comparability of data. This can be ensured by consistency in the
method of selection of the units for compilation of index numbers.
However, difficulties arise in the selection of commodities because the relative importance of
commodities keep on changing with the advancement of the society. More so, if the period is
quite long, these changes are quite significant both in the basket of production and the uses made
by people.
Base Period: It should be carefully selected because it is a point of reference in comparing various
data describing individual behaviour. The period should be normal i.e., one of the relative stability,
not affected by extraordinary events like war, famine, etc. It should be relatively recent because
we are more concerned with the changes with reference to the present and not with the distant
past. There are three variants of the base fixed, chain, and the average.
Selection of Weights: It is necessary to point out that each variable involved in composite index
should have a reasonable influence on the index, i.e., due consideration should be given to the
relative importance of each variable which relates to the purpose for which the index is to be
used. For example, in the computation of cost of living index, sugar cannot be given the same
importance as the cereals.
Use of Averages: Since we have to arrive at a single index number summarising a large amount
of information, it is easy to realise that average plays an important role in computing index
numbers. The geometric mean is better in averaging relatives, but for most of the indices arithmetic
mean is used because of its simplicity.
Choice of Variables: Index numbers are constructed with regard to price or quantity or any
other measure. We have to decide about the unit. For example, in price index numbers it is
necessary to decide whether to have wholesale or the retail prices. The choice would depend on
the purpose. Further, it is necessary to decide about the period to which such prices will be
related. There may be an average of price for certain time-period or the end of the period. The
former is normally preferred.
Selection of Formula: The question of selection of an appropriate formula arises, since different
types of indices give different values when applied to the same data. We will see different types
of indices to be used for construction succeedingly.

Notations: It is customary to let Pn(1), Pn(2), Pn(3) denote the prices during nth period for the first,
second and third commodity. The corresponding price during a base period are denoted by Po(1),
Po(2), Po(3), etc. With these notations the price of commodity j during period n can be indicated by
Pn( j). We can use the summation notation by summing over the superscripts j as follows:
k
 Pn (j) or  Pn( j )
j=1
We can omit the superscript altogether and write as Pn etc.
Relatives: One of the simplest examples of an index number is a price relative, which is the ratio
of the price of single commodity in a given period to its price in another period called the base
period or the reference period. It can be indicated as follows:

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18.4 STATISTICS

Pn
Price relative = P
o

It has to be expressed as a percentage, it is multiplied by 100

Pn
Price relative = P  100
o

There can be other relatives such as of quantities, volume of consumption, exports, etc. The
relatives in that case will be:
Qn
Quantity relative = Q
o

Similarly, there are value relatives:

Vn Pn Qn  Pn Qn 
Value relative = V     
o P o Q o  P o Q o 
When successive prices or quantities are taken, the relatives are called the link relative,
P1 P 2 P 3 P n
, , ,
P o P 1 P 2 P n1
When the above relatives are in respect to a fixed base period these are also called the chain relatives
with respect to this base or the relatives chained to the fixed base. They are in the form of :
P1 P 2 P 3 P n
, , ,
Po Po Po Po

Methods: We can state the broad heads as follows:

Methods

Simple Weighted

Aggregative Relative Aggregative Relative

In this method of computing a price index, we express the total of commodity prices in a given
year as a percentage of total commodity price in the base year. In symbols, we have
P n
Simple aggregative price index =  100
P o
where Pn is the sum of all commodity prices in the current year and Po is the sum of all
commodity prices in the base year.

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 INDEX NUMBERS 18.5

ILLUSTRATIONS:
Commodities 1998 1999 2000
Cheese (per 100 gms) 12.00 15.00 15.60
Egg (per piece) 3.00 3.60 3.30
Potato (per kg) 5.00 6.00 5.70
Aggregrate 20.00 24.60 24.60
Index 100 123 123

P n 24.60
Simple Aggregative Index for 1999 over 1998 =  P  20.00  100 123
o

P n 24.60
and for 2000 over 1998 =  P  100 20.00  100 123
o

The above method is easy to understand but it has a serious defect. It shows that the first
commodity exerts greater influence than the other two because the price of the first commodity
is higher than that of the other two. Further, if units are changed then the Index numbers will
also change. Students should independently calculate the Index number taking the price of eggs
per dozen i.e., ` 36, ` 43.20, ` 39.60 for the three years respectively. This is the major flaw in using
absolute quantities and not the relatives. Such price quotations become the concealed weights
which have no logical significance.

One way to rectify the drawbacks of a simple aggregative index is to construct a simple average
of relatives. Under it we invert the actual price for each variable into percentage of the base
period. These percentages are called relatives because they are relative to the value for the base
period. The index number is the average of all such relatives. One big advantage of price relatives
is that they are pure numbers. Price index number computed from relatives will remain the same
regardless of the units by which the prices are quoted. This method thus meets criterion of unit
test (discussed later). Also quantity index can be constructed for a group of variables that are
expressed in divergent units.

ILLUSTRATIONS:
In the proceeding example we will calculate relatives as follows:
Commodities 1998 1999 2000
A 100.0 125.0 130.0
B 100.0 120.0 110.0
C 100.0 120.0 114.0
Aggregate 300.0 365.0 354.0
Index 100.0 121.67 118.0

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18.6 STATISTICS

Inspite of some improvement, the above method has a flaw that it gives equal importance to each
of the relatives. This amounts to giving undue weight to a commodity which is used in a small
quantity because the relatives which have no regard to the absolute quantity will give weight
more than what is due from the quantity used. This defect can be remedied by the introduction
of an appropriate weighing system.

To meet the weakness of the simple or unweighted methods, we weigh the price of each
commodity by a suitable factor often taken as the quantity or the volume of the commodity sold
during the base year or some typical year. These indices can be classfied into broad groups:
(i) Weighted Aggregative Index.
(ii) Weighted Average of Relatives.
(i) Weighted Aggregative Index: Under this method we weigh the price of each commodity by a
suitable factor often taken as the quantity or value weight sold during the base year or the given
year or an average of some years. The choice of one or the other will depend on the importance
we want to give to a period besides the quantity used. The indices are usually calculated in
percentages. The various alternatives formulae in use are:
(The example has been given after the tests).
(a) Laspeyres’ Index: In this Index base year quantities are used as weights:
PnQ0
Laspeyres Index = × 100
P0Q0
(b) Paasche’s Index: In this Index current year quantities are used as weights:
P Q
Passche's Index = n n × 100
PoQn
(c) Methods based on some typical Period:
Pn Qt
Index  100 the subscript t stands for some typical period of years, the quantities of
Po Q t
which are used as weight
Note: * Indices are usually calculated as percentages using the given formulae
The Marshall-Edgeworth index uses this method by taking the average of the base year and
the current year
 Pn  Qo + Qn 
Marshall-Edgeworth Index = 100
 Po  Qo + Qn 
(d) Fisher’s ideal Price Index: This index is the geometric mean of Laspeyres’ and Paasche’s.
 Pn Q o  Pn Q n
Fisher's Index = 
 Po Q o  Po Q n ×100

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 INDEX NUMBERS 18.7

(ii) Weighted Average of Relative Method: To overcome the disadvantage of a simple average
of relative method, we can use weighted average of relative method. Generally weighted
arithmetic mean is used although the weighted geometric mean can also be used. The
weighted arithmetic mean of price relatives using base year value weights is represented by
Pn
  (Po Q o )
Po 
× 100 = 

 Po Q o
Example:
Price Relatives Value Weights Weighted Price Relatives
Commodity
Q. 1998 1999 2000 1998 1999 2000
Pn Pn Pn P0Q0 Pn Pn
P0Q0 P0Q0
P0 P0 P0 P0 P0
Butter 0.7239 100 101.1 118.7 72.39 73.19 85.93
Milk 0.2711 100 101.7 126.7 27.11 27.57 34.35
Eggs 0.7703 100 100.9 117.8 77.03 77.72 90.74
Fruits 4.6077 100 96.0 114.7 460.77 442.34 528.50
Vegetables 1.9500 100 84.0 93.6 195.00 163.80 182.52
832.30 784.62 922.04

Weighted Price Relative

784.62
For 1999 :  100 = 94.3
832.30
922.04
For 2000 :  100 = 110.8
832.30

So far we concentrated on a fixed base but it does not suit when conditions change quite fast. In
such a case the changing base for example, 1998 for 1999, and 1999 for 2000, and so on, may be
more suitable. If, however, it is desired to associate these relatives to a common base the results
may be chained. Thus, under this method the relatives of each year are first related to the preceding
year called the link relatives and then they are chained together by successive multiplication to
form a chain index.

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18.8 STATISTICS

The formula is:

Link relative of current year  Chain Index of the previous year
Chain Index =
100
Example:
The following are the index numbers by a chain base method:
Year Price Link Relatives Chain Indices
(1) (2) (3) (4)

1991 50 100 100

60 120
1992 60  100 = 120.0  100 = 120.0
50 100
62 103.3
1993 62  100 = 103.3  120 = 124.0
60 100
65 104.8
1994 65  100 = 104.8  124 = 129.9
62 100
70 107.7
1995 70  100 = 107.7  129.9 = 139.9
65 100
78 111.4
1996 78  100 = 111.4  139.9 = 155.8
70 100
82 105.1
1997 82  100 = 105.1  155.8 = 163.7
78 100
84 102.4
1998 84  100 = 102.4  163.7 = 167.7
82 100
88 104.8
1999 88  100 = 104.8  167.7 = 175.7
84 100
90 102.3
2000 90  100 = 102.3  175.7 = 179.7
88 100
You will notice that link relatives reveal annual changes with reference to the previous year. But
when they are chained, they change over to a fixed base from which they are chained, which in the
above example is the year 1991. The chain index is an unnecessary complication unless of course
where data for the whole period are not available or where commodity basket or the weights have
to be changed. The link relatives of the current year and chain index from a given base will give also
a fixed base index with the given base year as shown in the column 4 above.

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 INDEX NUMBERS 18.9

To measure and compare prices, we use price index numbers. When we want to measure and
compare quantities, we resort to Quantity Index Numbers. Though price indices are widely used
to measure the economic strength, Quantity indices are used as indicators of the level of output
in economy. To construct Quantity indices, we measure changes in quantities and weight them
using prices or values as weights. The various types of Quantity indices are:
1. Simple aggregate of quantities:

 Qn
This has the formula  Q
o

2. The simple average of quantity relatives:

 Qn
This can be expressed by the formula  Q o
N

3. Weighted aggregate Quantity indices:

 Q n Po
(i) With base year weight :  Q P (Laspeyre’s index)
o o

 Q n Pn
(ii) With current year weight :  Q P (Paasche’s index)
o n

 Q n Po  Q n Pn
(iii) Geometric mean of (i) and (ii) : 
 Q o Po  Q o Pn (Fisher’s Ideal)

Q 
 n Po Q o 
4. Base-year weighted average of quantity relatives. This has the formula  Qo 
 Po Q o

Note : Indices are usually calculated as percentages using the given formulae.

Value equals price multiplied by quantity. Thus a value index equals the total sum of the
values of a given year divided by the sum of the values of the base year, i.e.,

 Vn  Pn Q n

 Vo  P0 Q 0

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18.10 STATISTICS

So far we have studied various types of index numbers. However, they have certain limitations.
They are :
1. As the indices are constructed mostly from deliberate samples, chances of errors creeping in
cannot be always avoided.
2. Since index numbers are based on some selected items, they simply depict the broad trend and
not the real picture.
3. Since many methods are employed for constructing index numbers, the result gives different
values and this at times create confusion.
In spite of its limitations, index numbers are useful in the following areas :
1. Framing suitable policies in economics and business. They provide guidelines to make
decisions in measuring intelligence quotients, research etc.
2. They reveal trends and tendencies in making important conclusions in cyclical forces,
irregular forces, etc.
3. They are important in forecasting future economic activity. They are used in time series
analysis to study long-term trend, seasonal variations and cyclical developments.
4. Index numbers are very useful in deflating i.e., they are used to adjust the original data for
price changes and thus transform nominal wages into real wages.
5. Cost of living index numbers measure changes in the cost of living over a given period.

Sometimes a price index is used to measure the real values in economic time series data expressed
in monetary units. For example, GNP initially is calculated in current price so that the effect of
price changes over a period of time gets reflected in the data collected. Thereafter, to determine
how much the physical goods and services have grown over time, the effect of changes in price
over different values of GNP is excluded. The real economic growth in terms of constant prices
of the base year therefore is determined by deflating GNP values using price index.
Year Wholesale GNP Real
Price Index at Current Prices GNP
1970 113.1 7499 6630
1971 116.3 7935 6823
1972 121.2 8657 7143
1973 127.7 9323 7301

The formula for conversion can be stated as

Current Value
Deflated Value =
Price Index of the current year

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 INDEX NUMBERS 18.11

Base Price (P0)

or Current Value 
Current Price (Pn)

These refer to two technical points: (i) how the base period of the index may be shifted, (ii) how
two index covering different bases may be combined into single series by splicing.
Shifted Price Index
Year Original Price Index Shifted Price Index to base 1990
1980 100 71.4
1981 104 74.3
1982 106 75.7
1983 107 76.4
1984 110 78.6
1985 112 80.0
1986 115 82.1
1987 117 83.6
1988 125 89.3
1989 131 93.6
1990 140 100.0
1991 147 105.0
The formula used is,
Original Price Index
Shifted Price Index =  100
Price Index of the year on which it has to be shifted

Splicing two sets of price index numbers covering different periods of time is usually required
when there is a major change in quantity weights. It may also be necessary on account of a new
method of calculation or the inclusion of new commodity in the index.

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18.12 STATISTICS

Splicing Two Index Number Series

Year Old Price Revised Price Spliced Price
Index Index Index
[1990 = 100] [1995 = 100] [1995 = 100]
1990 100.0 87.6
1991 102.3 89.6
1992 105.3 92.2
1993 107.6 94.2
1994 111.9 98.0
1995 114.2 100.0 100.0
1996 102.5 102.5
1997 106.4 106.4
1998 108.3 108.3
1999 111.7 111.7
2000 117.8 117.8
You will notice that the old series upto 1994 has to be converted shifting to the base. 1995 i.e,
114.2 to have a continuous series, even when the two parts have different weights

There are four tests:

(i) Unit Test: This test requires that the formula should be independent of the unit in which or
for which prices and quantities are quoted. Except for the simple (unweighted) aggregative
index all other formulae satisfy this test.
(ii) Time Reversal Test: It is a test to determine whether a given method will work both ways in
time, forward and backward. The test provides that the formula for calculating the index
number should be such that two ratios, the current on the base and the base on the current
should multiply into unity. In other words, the two indices should be reciprocals of each
other. Symbolically,
P01  P10 = 1
where P01 is the index for time 1 on 0 and P10 is the index for time 0 on 1.
You will notice that Laspeyres’ method and Paasche’s method do not satisfy this test, but Fisher’s
Ideal Formula does.
While selecting an appropriate index formula, the Time Reversal Test and the Factor Reversal
test are considered necessary in testing the consistency.

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 INDEX NUMBERS 18.13

Laspeyres:
P1Q0 ,
P0Q1
P01 = P10 =
P0Q0 P1Q1
P1Q0 P0Q1
P01  P10 =  1
P0Q0 P1Q1
Paasche’s
P1Q1 ,
P0Q0
P01 = P10 =
P0Q1 P1Q0
P1Q1 P0Q0
 P01  P10 =  1
P0Q1 P1Q0
Fisher’s :
P1Q0 P1Q1 P0Q1 P0Q0
P01 = x P10 = 
P0Q0 P0Q1 P1Q1 P1Q0
P1Q0 P1Q1 P0Q1 P0Q0
 P01  P10 =  x  =1
P0Q0 P0Q1 P1Q1 P1Q0

(iii) Factor Reversal Test: This holds when the product of price index and the quantity index
P 1 Q1
should be equal to the corresponding value index, i.e., P Q
0 0
Symbolically: P01  Q01 = V01

 P1Q 0  P1 Q 1 Q1 P0 Q1 P 1

Fishers’ P01 =  Q01= 
 P0 Q 0  P0 Q 1  Q 0 P0  Q 0 P1

 P1Q 0 P 1 Q 1  Q 1P0  Q 1P1  P1Q 1  P1 Q 1

P01 × Q01 =     
 P0 Q 0  P0 Q 1  Q 0 P0  Q 0 P 1  P 0 Q 0  P0 Q 0

 P1 Q 1
=
 P0 Q 0
Thus Fisher’s Index satisfies Factor Reversal test. Because Fisher’s Index number satisfies
both the tests in (ii) and (iii), it is called an Ideal Index Number.
(iv) Circular Test: It is concerned with the measurement of price changes over a period of years,
when it is desirable to shift the base. For example, if the 1970 index with base 1965 is 200 and 1965
index with base 1960 is 150, the index 1970 on base 1960 will be 300. This property therefore
enables us to adjust the index values from period to period without referring each time to the
original base. The test of this shiftability of base is called the circular test.
This test is not met by Laspeyres, or Paasche’s or the Fisher’s ideal index. The simple geometric
mean of price relatives and the weighted aggregative with fixed weights meet this test.

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18.14 STATISTICS

Example: Compute Fisher’s Ideal Index from the following data:

Base Year Current Year
Commodities Price Quantity Price Quantity
A 4 3 6 2
B 5 4 6 4
C 7 2 9 2
D 2 3 1 5
Show how it satisfies the time and factor reversal tests.
Solution:
Commodities P0 Q0 P1 Q1 P0Q0 P1Q0 P0Q1 P1Q1
A 4 3 6 2 12 18 8 12
B 5 4 6 4 20 24 20 24
C 7 2 9 2 14 18 14 18
D 2 3 1 5 6 3 10 5
52 63 52 59

P1Q0 P1Q1 63 59
Fisher’s Ideal Index: P01 =   100 =   100
P0Q0 P0Q1 52 52

= 1.375  100 = 1.172  100 = 117.3

Time Reversal Test:

63 59 52 52
P01  P10 =    = 1 = 1
52 52 59 63
 Time Reversal Test is satisfied.
Factor Reversal Test:
63 59 52 52 59 59 59
P01  Q01 =    =  =
52 52 59 63 52 52 52
P1Q1 59
Since, is also equal to , the Factor Reversal Test is satisfied.
P0Q0 52

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 INDEX NUMBERS 18.15

Concept Insight
Stock Market Index: It represents the entire stock market. It shows the changes taking place
in the stock market. Movement of index is also an indication of average returns received by
the investors. With the help of an index, it is easy for an investor to compare performance as
it can be used as a benchmark, for e.g. a simple comparison of the stock and the index can
be undertaken to find out the feasibility of holding a particular stock.
Each stock exchange has an index. For instance, in India, it is Sensex of BSE and Nifty
of NSE. On the other hand, in outside India, popular indexes are Dow Jones, NASDAQ,
FTSE etc.
(a) Bombay Stock Exchange Limited: It is the oldest stock exchange in Asia and was
established as “The Native Share & Stock Brokers Association” in 1875. The Securities
Contract (Regulation) Act, 1956 gives permanent recognition to Bombay Stock Exchange
in 1956. BSE became the first stock exchange in India to obtain such permission from
the Government under the Act. One of the Index as BSE Sensex which is basket of 30
constituent stocks. The base year of BSE SENSEX is 1978-79 and the base value is 100
which has grown over the years and quoted at about 592 times of base index as on date.
As the oldest Index in the country, it provides the time series data over a fairly long
period of time ( from 1979 onward).
(b) National Stock Exchange: NSE was incorporated in 1992. It was recognized as a
stock exchange by SEBI in April 1993 and commenced operations in 1994.NIFTY50 is
a diversified 50 stocks Index of 13 sectors of the economy. The base period of NIFTY
50 Index is 3 November 1995 and base value is 1000 which has grown over years and
quoted at 177 times as on date.

Computation of Index
Following steps are involved in calculation of index on a particular date:
• Calculate market capitalization of each individual company comprising the index.
• Calculate the total market capitalization by adding the individual market capitalization
of all companies in the index.
• Computing index of next day requires the index value and the total market capitalization
of the previous day and is computed as follows:
Total market capitalisation for current day
IndexValue=Index on Previous Day ×
Total market capitalisation for previous day

• It should also be noted that Indices may also be calculated using the price weighted
method. Here, the share price of the constituent companies forms the weight. However,
almost all equity indices worldwide are calculated using the market capitalization
weighted method.

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18.16 STATISTICS

• It is very important to note that constituents’ companies does not remain the same.
Hence , it may be possible the stocks of the company constituting index at the time of
index inspection , may not be aprt of index as on date and new companies stock may
have replaced them.

CPI- Consumer Price Index/ Cost of living Index or Retail Price Index is the Index which
measures the effect of change in prices of basket of goods and services on the purchasing
power of specific class of consumer during any current period w.r.t to some base period.
WPI- Whole Sale Price Index - The WPI measures the relative changes in prices of commodities
traded in wholesale market.

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.17

 An index number is a ratio or an average of ratios expressed as a percentage, Two or

more time periods are involved, one of which is the base time period.
 Issues Involved in index numbers
(a) Selection of Data
(b) Base period
(c) Selection of Weights:
(d) Use of Averages:
(e) Choice of Variables
 Construction of Index Number
Price Index numbers

(a) Simple aggregative price index =

P n
×100
P o

(b) Laspeyres’ Index: In this Index base year quantities are used as weights:

P Q n o
×100
Laspeyres Index =
P Q o o

(c) Paasche’s Index: In this Index current year quantities are used as weights:

P Qn n
×100
Passche’s Index =
P Qo n

(d) The Marshall-Edgeworth index uses this method by taking the average of the base year
and the current year

 P (Qn o +Q n )
×100
Marshall-Edgeworth Index =
 P (Qo o +Q n )

(e) Fisher’s ideal Price Index: This index is the geometric mean of Laspeyres’ and
Paasche’s.

P Q × P Q
n o n n
×100
Fisher’s Index =
P Q P Q
o o o n

P n
×(Po Q o )
(g) Weighted Average of Relative Method:
Po
×100=
P Q
n o
×100
 Po Q o P Q
o o

Link relative of current year × Chain Index of the previous year

(h) Chain Index =
100

© The Institute of Chartered Accountants of India

18.18 STATISTICS

Quantity Index Numbers

Q n
Simple aggregate of quantities:

Q o

Q n

 The simple average of quantity relatives:  Q o

N
 Weighted aggregate quantity indices:

Q P n o
(i) With base year weight : (Laspeyre’s index)
Q P o o

Q P n n
(ii) With current year weight : (Paasche’s index)
Q P o n

Q P × Q P
n o n n
(iii)Geometric mean of (i) and (ii) : (Fisher’s Ideal)
Q P Q P
o o o n

 Qn 
  Q P Q o
o o 

 Base-year weighted average of quantity relatives. This has the formula
P Q o o

 Value Indices

Vn
=
P Q
n n

Vo P Q
o o

Current Value
 Deflated Value =
Price Index of the current year

Base Price (P0 )

or Current Value x Base Price (P0)
Current Price (Pn )

Original Price Index

 Shifted Price Index = ×100
Price Index of the year on which it has to be shifted

 Test of Adequacy
(1) Unit test (2) Time reversal Test
(3) Factor reversal test (4) Circular Test

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.19

Choose the most appropriate option (a) (b) (c) or (d).

1. A series of numerical figures which show the relative position is called
a) index number b) relative number c) absolute number d) none
2. Index number for the base period is always taken as
a) 200 b) 50 c) 1 d) 100
3. _________ play a very important part in the construction of index numbers.
a) weights b) classes c) estimations d) none
4. ________ is particularly suitable for the construction of index numbers.
a) H.M. b) A.M. c) G.M. d) none
5. Index numbers show _________ changes rather than absolute amounts of change.
a) relative b) percentage c) both d) none
6. The ________ makes index numbers time-reversible.
a) A.M. b) G.M. c) H.M. d) none
7. Price relative is equal to

Price in the given year ×100 Price in the year base year × 100
a) b)
Price in the base year Price in the given year
c) Price in the given year  100 d) Price in the base year  100
8. Index number is equal to
a) sum of price relatives b) average of the price relatives
c) product of price relative d) none
9. The ________ of group indices gives the General Index
a) H.M. b) G.M. c) A.M. d) none
10. Circular Test is one of the tests of
a) index numbers b) hypothesis c) both d) none
11. ___________ is an extension of time reversal test
a) Factor Reversal test b) Circular test
c) both d) none
12. Weighted G.M. of relative formula satisfy ________test
a) Time Reversal Test b) Circular test
c) Factor Reversal Test d) none
13. Factor Reversal test is satisfied by
a) Fisher’s Ideal Index b) Laspeyres Index
c) Paasches Index d) none

© The Institute of Chartered Accountants of India

18.20 STATISTICS

14. Laspeyre's formula does not satisfy

a) Factor Reversal Test b) Time Reversal Test
c) Circular Test d) all the above
15. A ratio or an average of ratios expressed as a percentage is called
a) a relative number b) an absolute number
c) an index number d) none
16. The value at the base time period serves as the standard point of comparison
a) false b) true c) both d) none
17. An index time series is a list of _______ numbers for two or more periods of time
a) index b) absolute c) relative d) none
18. Index numbers are often constructed from the
a) frequency b) class c) sample d) none
19. ________ is a point of reference in comparing various data describing individual behaviour.
a) Sample b) Base period c) Estimation d) none
20. The ratio of price of single commodity in a given period to its price in the preceding year
price is called the
(a) base period (b) price ratio (c) relative price (d) none
Sum of all commodity prices in the current year × 100
21. Sum of all commodity prices in the base year is
(a) Relative Price Index (b) Simple Aggregative Price Index
(c) both (d) none
22. Chain index is equal to
link relative of current year × chain index of the current year
(a)
100
link relative of previous year × chain index of the current year
(b)
100
link relative of current year × chain index of the previous year
(c)
100
link relative of previous year × chain index of the previous year
(d)
100
23. P01 is the index for time
(a) 1 on 0 (b) 0 on 1 (c) 1 on 1 (d) 0 on 0
24. P10 is the index for time
(a) 1 on 0 (b) 0 on 1 (c) 1 on 1 (d) 0 on 0

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.21

25. When the product of price index and the quantity index is equal to the corresponding value
index then the test that holds is
(a) Unit Test (b) Time Reversal Test
(c) Factor Reversal Test (d) none holds
26. The formula should be independent of the unit in which or for which price and quantities
are quoted in
(a) Unit Test (b) Time Reversal Test
(c) Factor Reversal Test (d) none
27. Laspeyre's method and Paasche’s method do not satisfy
(a) Unit Test (b) Time Reversal Test
(c) Factor Reversal Test (d) b & c
28. The purpose determines the type of index number to use
(a) yes (b) no (c) may be (d) may not be
29. The index number is a special type of average
(a) false (b) true (c) both (d) none
30. The choice of suitable base period is at best temporary solution
(a) true (b) false (c) both (d) none
31. Fisher’s Ideal Formula for calculating index numbers satisfies the _______ tests
(a) Unit Test (b) Factor Reversal Test
(c) both (d) none
32. Fisher’s Ideal Formula dose not satisfy _________ test
(a) Unit Test (b) Circular Test (c) Time Reversal Test (d) none
33. _________________ satisfies circular test
a) G.M. of price relatives or the weighted aggregate with fixed weights
b) A.M. of price relatives or the weighted aggregate with fixed weights
c) H.M. of price relatives or the weighted aggregate with fixed weights
d) none
34. Laspeyre's and Paasche’s method _________ time reversal test
(a) satisfy (b) do not satisfy (c) are (d) are not
35. There is no such thing as unweighted index numbers
(a) false (b) true (c) both (d) none
36. Theoretically, G.M. is the best average in the construction of index numbers but in practice,
mostly the A.M. is used
(a) false (b) true (c) both (d) none

© The Institute of Chartered Accountants of India

18.22 STATISTICS

37. Laspeyre’s or Paasche’s or the Fisher’s ideal index do not satisfy

(a) Time Reversal Test (b) Unit Test
(c) Circular Test (d) none
38. ___________ is concerned with the measurement of price changes over a period of years,
when it is desirable to shift the base
(a) Unit Test (b) Circular Test
(c) Time Reversal Test (d) none
39. The test of shifting the base is called
(a) Unit Test (b) Time Reversal Test
(c) Circular Test (d) none
40. The formula for conversion to current value
Price Index of the current year
(a) Deflated value =
previous value
current value
(b) Deflated value =
Price Index of the current year
Price Index of the previous year
(c) Deflated value =
previous value
Price Index of the previous year
(d) Deflated value =
previous value
Original Price ×100
41. Shifted price Index =
Price Index of the year on which it has to be shifted
(a) True (b) false (c) both (d) none
42. The number of test of Adequacy is
(a) 2 (b) 5 (c) 3 (d) 4
43. We use price index numbers
(a) To measure and compare prices (b) to measure prices
(c) to compare prices(d) none
44. Simple aggregate of quantities is a type of
(a) Quantity control (b) Quantity indices
(c) both (d) none

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.23

Exercise
1. (a) 2. (d) 3. (a) 4. (c) 5. (b) 6. (b) 7. (a) 8. (b)
9. (c) 10. (a) 11. (b) 12. (a) 13. (a) 14. (d) 15. (c) 16. (b)
17. (a) 18. (c) 19. (b) 20. (c) 21. (b) 22. (c) 23. (a) 24. (b)
25. (c) 26. (a) 27. (d) 28. (a) 29. (b) 30. (a) 31. (c) 32. (b)
33. (a) 34. (b) 35. (a) 36. (b) 37. (c) 38. (b) 39. (c) 40. (b)
41. (a) 42. (d) 43. (a) 44. (b)

1. Each of the following statements is either True or False write your choice of the answer by
writing T for True
(a) Index Numbers are the signs and guideposts along the business highway that indicate
to the businessman how he should drive or manage.
(b) “For Construction index number, the best method on theoretical ground is not the best
method from practical point of view”.
(c) Weighting index numbers makes them less representative.
(d) Fisher’s index number is not an ideal index number.
2. Each of the following statements is either True or False. Write your choice of the answer by
writing F for false.
(a) Geometric mean is the most appropriate average to be used for constructing an index
number.
(b) Weighted average of relatives and weighted aggregative methods render the same result.
(c) “Fisher’s Ideal Index Number is a compromise between two well known indices – not a
right compromise, economically speaking”.
(d) “Like all statistical tools, index numbers must be used with great caution”.
3. The best average for constructing an index numbers is
(a) Arithmetic Mean (b) Harmonic Mean
(c) Geometric Mean (d) None of these.
4. The time reversal test is satisfied by
(a) Fisher’s index number. (b) Paasche’s index number.
(c) Laspeyre’s index number. (d) None of these.

© The Institute of Chartered Accountants of India

18.24 STATISTICS

5. The factor reversal test is satisfied by

(a) Simple aggregative index number. (b) Paasche’s index number.
(c) Laspeyre’s index number. (d) Fisher’s index
6. The circular test is satisfied by
(a) Fisher’s index number. (b) Paasche’s index number.
(c) Laspeyre’s index number. (d) Simple GM price relative.
7. Fisher’s index number is based on
(a) The Arithmetic mean of Laspeyre’s and Paasche’s index numbers.
(b) The Median of Laspeyre’s and Paasche’s index numbers.
(c) the Mode of Laspeyre’s and Paasche’s index numbers.
(d) None of these.
8. Paasche index is based on
(a) Base year quantities. (b) Current year quantities.
(c) Average of current and base year. (d) None of these.
9. Fisher’s ideal index number is
(a) The Median of Laspeyre’s and Paasche’s index numbers
(b) The Arithmetic Mean of Laspeyre’s and Paasche’s index numbers
(c) The Geometric Mean of Laspeyre’s and Paasche’s index numbers
(d) None of these.
10. Price-relative is expressed in term of
Pn Po
(a) P  (b) P 
Po Pn

Pn Po
(c) P   100 (d) P   100
Po Pn

11. Paasehe’s index number is expressed in terms of :

 Pn q n  Po q o
(a)  P q (b)  P q
o n n n

 Pn q n  Pn q o
(c)  P q  100 (d)  P q  100
o n o o

12. Time reversal Test is satisfied by following index number formula is

(a) Laspeyre’s Index number.

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.25

(b) Simple Arithmetic Mean of price relative formula

(c) Marshall-Edge worth formula.
(d) None of these.
13. Cost of Living Index number (C. L. I.) is expressed in terms of :
 Pn q o  Pn q n
(a)  P q  100 (b)  P q
o o o o

 Po q n
(c)  P q  100 (d) None of these.
n n

14. If the ratio between Laspeyre’s index number and Paasche’s Index number is 28 : 27. Then
the missing figure in the following table P is :
Commodity Base Year Current Year
Price Quantity Price Quantity
X L 10 2 5
Y L 5 P 2
(a) 7 (b) 4 (c) 3 (d) 9
15. If the prices of all commodities in a place have increased 1.25 times in comparison to the
base period, the index number of prices of that place now is
(a) 125 (b) 150 (c) 225 (d) None of these.
16. If the index number of prices at a place in 1994 is 250 with 1984 as base year, then the prices
have increased on average by
(a) 250% (b) 150% (c) 350% (d) None of these.
17. If the prices of all commodities in a place have decreased 35% over the base period prices,
then the index number of prices of that place is now
(a) 35 (b) 135 (c) 65 (d) None of these.

18. Link relative index number is expressed for period n is

P P
n 0
(a) P (b) P
n1 n 1
P
n  100
(c) P (d) None of these.
n 1
19. Fisher’s Ideal Index number is expressed in terms of :
(a) (Pon)F = Laspeyre's Index ×Paasche's Index 
(b) (Pon)F = Laspeyre’s Index X Paasche’s Index

© The Institute of Chartered Accountants of India

18.26 STATISTICS

(c) (Pon)F = Marshall Edge worth Index ×Paasche's

(d) None of these.
20. Factor Reversal Test According to Fisher is P01 × Q01=
 Po q o  Pn q n
(a) (b)
 Pn q n  Po q o

 Po q n
(c) (d) None of these.
 Pn q n
21. Marshall-edge worth Index formula after interchange of p and q is expressed in terms of :
 q n  p0  p n  P n (q 0  q n )
(a) (b)  qP (q  q )
 q 0  p0  p n  0 0 n

P 0 (q 0  q n )
(c)  P (P  P ) (d) None of these.
n 0 n

22. If  Pnqn = 249,  Poqo = 150, Paasche’s Index Number = 150 and Drobiseh and Bowely’s
Index number = 145, then the Fisher’s Ideal Index Number is
(a) 75 (b) 60 (c) 145.97 (d) None of these.
23. Consumer Price index number for the year 1957 was 313 with 1940 as the base year. The
Average Monthly wages in 1957 of the workers into factory be ` 160/- their real wages is
(a) ` 48.40 (b) ` 51.12 (c) ` 40.30 (d) None of these.
24. If  Poqo = 3500,  Pnqo = 3850, then the Cost of living Index (C.L.I.) for 1950 w.r. to base 1960
is
(a) 110 (b) 90 (c) 100 (d) None of these.
25. From the following table by the method of relatives using Arithmetic mean the price Index
number is
Commodity Wheat Milk Fish Sugar
Base Price 5 8 25 6
Current Price 7 10 32 12
(a) 140.35 (b) 148.25 (c) 140.75 (d) None of these.
From the Q.No. 26 to 29 each of the following statements is either True or False with your
choice of the answer by writing F for False.
26. (a) Base year quantities are taken as weights in Laspeyre’s price Index number.
(b) Fisher’s ideal index is equal to the Arithmetic mean of Laspeyre’s and Paasche’s index
numbers.

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.27

(c) Laspeyre’s index number formula does not satisfy time reversal test.
(d) None of these.
27. (a) Current year quantities are taken as weights in Paasche’s price index number.
(b) Edge worth Marshall’s index number formula satisfies Time, Reversal Test.
(c) The Arithmetic mean of Laspeyre’s and Paasche’s index numbers is called Bowely’s
index numbers.
(d) None of these.
28. (a) Current year prices are taken as weights in Paasche’s quantity index number.
(b) Fisher’s Ideal Index formula satisfies factor Reversal Test.
(c) The sum of the quantities of the base period and current period is taken as weights in
Laspeyre’s index number.
(d) None of these.
29. (a) Simple Aggregative and simple Geometric mean of price relatives formula satisfy circular
Test.
(b) Base year prices are taken as weights in Laspeyre’s quantity index numbers.
(c) Fisher’s Ideal Index formula obeys time reversal and factor reversal tests.
(d) None of these.
30. In 1980, the net monthly income of the employee was ` 800/- p. m. The consumer price
index number was 160 in 1980. It rises to 200 in 1984. If he has to be rightly compensated.
The additional D. A. to be paid to the employee is
(a) ` 175/- (b) ` 185/- (c) ` 200/- (d) ` 125.
31. The simple Aggregative formula and weighted aggregative formula satisfy is
(a) Factor Reversal Test (b) Circular Test
(c) Unit Test (d) None of these.
32. “Fisher’s Ideal Index is the only formula which satisfies”
(a) Time Reversal Test (b) Circular Test
(c) Factor Reversal Test (d) a & c.
33. “Neither Laspeyre’s formula nor Paasche’s formula obeys” :
(a) Time Reversal and factor Reversal Tests of index numbers.
(b) Unit Test and circular Tests of index number.
(c) Time Reversal and Unit Test of index number.
(d) None of these.
34. Bowley's index number is 150. Fisher's index number is 149.95. Paasche's index number is
(a) 146.13 (b) 154 (c) 148 (d) 156

© The Institute of Chartered Accountants of India

18.28 STATISTICS

35. With the base year 1960 the C. L. I. in 1972 stood at 250. x was getting a monthly Salary of `
500 in 1960 and ` 750 in 1972. In 1972 to maintain his standard of living in 1960 x has to
receive as extra allowances of
(a) ` 600/- (b) ` 500/- (c) ` 300/- (d) none of these.
36. From the following data base year :-
Commodity Base Year Current Year
Price Quantity Price Quantity
A 4 3 6 2
B 5 4 6 4
C 7 2 9 2
D 2 3 1 5
Fisher’s Ideal Index is
(a) 117.3 (b) 115.43 (c) 118.35 (d) 116.48
37. Which statement is False?
(a) The choice of suitable base period is at best a temporary solution.
(b) The index number is a special type of average.
(c) Those is no such thing as unweighted index numbers.
(d) Theoretically, geometric mean is the best average in the construction of index numbers
but in practice, mostly the arithmetic mean is used.
38. Factor Reversal Test is expressed in terms of
 P1 Q 1  P1 Q 1  P1 Q 1
(a)  P Q (b)  P Q   P Q
0 0 0 0 0 1

 P1Q 1  Q 1P0  P1Q 1

(c)  Q P (d)  Q P   Q P
0 1 0 0 0 1

39. Circular Test is satisfied by

(a) Laspeyre’s Index number.
(b) Paasche’s Index number
(c) The simple geometric mean of price relatives and the weighted aggregative with fixed
weights.
(d) None of these.

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.29

40. From the following data for the 5 groups combined

Group Weight Index Number
Food 35 425
Cloth 15 235
Power & Fuel 20 215
Rent & Rates 8 115
Miscellaneous 22 150
The general Index number is
(a) 270 (b) 269.2 (c) 268.5 (d) 272.5
41. From the following data with 1966 as base year
Commodity Quantity Units Values (`)
A 100 500
B 80 320
C 60 150
D 30 360
The price per unit of commodity A in 1966 is
(a) ` 5 (b) ` 6 (c) ` 4 (d) ` 12
42. The index number in whole sale prices is 152 for August 1999 compared to August 1998.
During the year there is net increase in prices of whole sale commodities to the extent of
(a) 45% (b) 35% (c) 52% (d) 48%
43. The value Index is expressed in terms of
 P1Q1  P1 Q 1
(a)  100 (b)  P Q
 P0 Q 0 0 0

 P0 Q 0  P0 Q 1   P1 Q 1
(c)  100 (d)  P Q   P Q
 P1Q1 0 0 1 0

44. Purchasing Power of Money is

(a) Reciprocal of price index number. (b) Equal to price index number.
(c) Unequal to price index number. (d) None of these.
45. The price level of a country in a certain year has increased 25% over the base period. The
index number is
(a) 25 (b) 125 (c) 225 (d) 2500

© The Institute of Chartered Accountants of India

18.30 STATISTICS

46. The index number of prices at a place in 1998 is 355 with 1991 as base. This means
(a) There has been on the average a 255% increase in prices.
(b) There has been on the average a 355% increase in price.
(c) There has been on the average a 250% increase in price.
(d) None of these.
47. If the price of all commodities in a place have increased 1.25 times in comparison to the base
period prices, then the index number of prices for the place is now
(a) 100 (b) 125 (c) 225 (d) None of the above.
48. The wholesale price index number or agricultural commodities in a given region at a given
date is 280. The percentage increase in prices of agricultural commodities over the base year
is :
(a) 380 (b) 280 (c) 180 (d) 80
49. If now the prices of all the commodities in a place have been decreased by 35% over the base
period prices, then the index number of prices for the place is now (index number of prices
of base period = 100)
(a) 100 (b) 135 (c) 65 (d) None of these.
50. From the data given below
Commodity Price Relative Weight
A 125 5
B 67 2
C 250 3
Then the suitable index number is
(a) 150.9 (b) 155.8 (c) 145.8 (d) None of these.
51. Bowley’s Index number is expressed in the form of :
Laspeyre's index + Paasche's index Laspeyre's index × Paasche's index
(a) (b)
2 2
Laspeyre's index - Paasche's index
(c) (d) None of these.
2
52. From the following data
Commodity Base Price Current Price
Rice 35 42
Wheat 30 35
Pulse 40 38
Fish 107 120

© The Institute of Chartered Accountants of India

 INDEX NUMBERS 18.31

The simple Aggregative Index is

(a) 115.8 (b) 110.8 (c) 112.5 (d) 113.4
53. With regard to Laspeyre’s and Paasche’s price index numbers, it is maintained that “If the
prices of all the goods change in the same ratio, the two indices will be equal for them the
weighting system is irrelevant; or if the quantities of all the goods change in the same ratio,
they will be equal, for them the two weighting systems are the same relatively”. Then the
above statements satisfy.
(a) Laspeyre’s Price index  Paasche’s Price Index.
(b) Laspeyre’s Price Index = Paasche’s Price Index.
(c) Laspeyre’s Price Index may be equal Paasche’s Price Index.
(d) None of these.
54. The quantity Index number using Fisher’s formula satisfies :
(a) Unit Test (b) Factor Reversal Test.
(c) Time Reversal Test. (d) All the three
55. For constructing consumer price Index is used :
(a) Marshall Edge worth Method. (b) Paasche’s Method.
(c) Dorbish and Bowley’s Method. (d) Laspeyre’s Method.
56. The cost of living Index (C.L.I.) is always :
(a) Weighted index (b) Price Index.
(c) Quantity Index. (d) None of these.
57. The Time Reversal Test is not satisfied to :
(a) Fisher’s ideal Index. (b) Marshall Edge worth Method.
(c) Laspeyre’s and Paasche Method. (d) None of these.
58. Given below are the data on prices of some consumer goods and the weights attached to the
various items. Compute price index number for the year 1985 (Base 1984 = 100)
Items Unit 1984 1985 Weight
Wheat Kg. 0.50 0.75 2
Milk Litre 0.60 0.75 5
Egg Dozen 2.00 2.40 4
Sugar Kg. 1.80 2.10 8
Shoes Pair 8.00 10.00 1
Then weighted average of price Relative Index is :
(a) 125.43 (b) 123.3 (c) 124.53 (d) 124.52

© The Institute of Chartered Accountants of India

18.32 STATISTICS

59. The Factor Reversal Test is as represented symbolically is :

 P1Q1
(a) P01 × Q01  (b) I01 × I10
 P0 Q 0

 P0 Q 0  P1 Q 1  P0 Q 1
(c)  P Q (d) 
1 1  P0 Q 0  Q 10 P0

60. If the 1970 index with base 1965 is 200 and 1965 index with base 1960 is 150, the index 1970
on base 1960 will be :
(a) 700 (b) 300 (c) 500 (d) 600
61. Circular Test is not met by :
(a) The simple Geometric mean of price relatives.
(b) The weighted aggregative with fixed weights.
(c) Laspeyre’s or Paasche’s or the fisher’s Ideal index.
(d) None of these.
62. From the following data
Commodity Base Year Current Year
Price Quantity Price Quantity
A 4 3 6 2
B 5 4 6 4
C 7 2 9 2
D 2 3 1 5
Then the value ratio is:
59 49 41 47
(a) (b) (c) (d)
52 47 53 53
63. The value index is equal to :
(a) The total sum of the values of a given year multiplied by the sum of the values of the
base year.
(b) The total sum of the values of a given year Divided by the sum of the values of the base
year.
(c) The total sum of the values of a given year plus by the sum of the values of the base
year.
(d) None of these.

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 INDEX NUMBERS 18.33

64. Time Reversal Test is represented symbolically by :

(a) P01 x P10 (b) P01 x P10 = 1
(c) P01 x P10  1 (d) None of these.
65. In 1996 the average price of a commodity was 20% more than in 1995 but 20% less than in
1994; and more over it was 50% more than in 1997 to price relatives using 1995 as base (1995
price relative 100). Reduce the data is :
(a) 150, 100, 120, 80 for (1994–97) (b) 135, 100, 125, 87 for (1994–97)
(c) 140, 100, 120, 80 for (1994–97) (d) None of these.
66. From the following data
Commodities Base Year Current Year
1922 1934
Price (`) Price
A 6 10
B 2 2
C 4 6
D 11 12
E 8 12
The price index number for the year 1934 is :
(a) 140 (b) 145 (c) 147 (d) None of these.
67. From the following data
Commodities Base Price Current Price
1964 1968
Rice 36 54
Pulse 30 50
Fish 130 155
Potato 40 35
Oil 110 110
The index number by unweighted methods :
(a) 116.8 (b) 117.25 (c) 115.35 (d) 119.37
68. The Bowley’s Price index number is represented in terms of :
(a) A.M. of Laspeyre’s and Paasche’s Price index number.
(b) G.M. of Laspeyre’s and Paasche’s Price index number.
(c) A.M. of Laspeyre’s and Walsh’s price index number.
(d) None of these.

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18.34 STATISTICS

69. Fisher’s price index number equal is :

(a) G.M. of Kelly’s price index number and Paasche’s price index number.
(b) G.M. of Laspeyre’s and Paasche’s Price index number.
(c) G.M. of Bowley’s price index number and Paasche’s price index number.
(d) None of these.
70. The price index number using simple G.M. of the n relatives is given by :
1 Pn 1 Pn
(a) log Ion = 2-
n
 log P o
(b) log Ion = 2+
n
 log P o

1 Pn
(c) log Ion =
2n
 log P o
(d) None of these.

71. The price of a number of commodities are given below in the current year 1975 and base
year 1970.
Commodities A B C D E F
Base Price 45 60 20 50 85 120
Current Price 55 70 30 75 90 130
For 1975 with base 1970 by the Method of price relatives using Geometrical mean, the price
index is :
(a) 125.3 (b) 124.3 (c) 128.8 (d) None of these.
72. From the following data
Group A B C D E F
Group Index 120 132 98 115 108 95
Weight 6 3 4 2 1 4
The general Index I is given by :
(a) 111.3 (b) 113.45 (c) 117.25 (d) 114.75
73. The price of a commodity increases from ` 5 per unit in 1990 to ` 7.50 per unit in 1995 and the
quantity consumed decreases from 120 units in 1990 to 90 units in 1995. The price and quantity
in 1995 are 150% and 75% respectively of the corresponding price and quantity in 1990.
Therefore, the product of the price ratio and quantity ratio is :
(a) 1.8 (b) 1.125 (c) 1.75 (d) None of these.
74. Test whether the index number due to Walsh given by :
 P1 Q 0 Q 1
I=  100 Satisfies is :-
 P0 Q 0 Q 1
(a) Time reversal Test. (b) Factor reversal Test.
(c) Circular Test. (d) None of these.

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 INDEX NUMBERS 18.35

75. From the following data

Group Weight Index Number
Base : 1952–53 = 100
Food 50 241
Clothing 2 21
Fuel and Light 3 204
Rent 16 256
Miscellaneous 29 179
The Cost of living index numbers is :
(a) 224.5 (b) 223.91 (c) 225.32 (d) None of these.
76. Consumer price index number goes up from 110 to 200 and the Salary of a worker is also
raised from ` 325 to ` 500. Therefore, in real terms, to maintain his previous standard of
living he should get an additional amount of :
(a) ` 85 (b) ` 90.91 (c) ` 98.25 (d) None of these.
77. The prices of a commodity in the year 1975 and 1980 were 25 and 30 respectively taking 1980
as base year the price relative is :
(a) 109.78 (b) 110.25 (c) 113.25 (d) None of these.
78. The average price of certain commodities in 1980 was ` 60 and the average price of the same
commodities in 1982 was ` 120. Therefore, the increase in 1982 on the basis of 1980 was
100%. 80. The decrease in 1980 with 1982 as base is: using 1982, comment on the above
statement is :
(a) The price in 1980 decreases by 60% using 1982 as base.
(b) The price in 1980 decreases by 50% using 1982 as base.
(c) The price in 1980 decreases by 90% using 1982 as base.
(d) None of these.
79. Cost of Living Index (C.L.I.) numbers are also used to find real wages by the process of
(a) Deflating of Index number. (b) Splicing of Index number.
(c) Base shifting. (d) None of these.
80. From the following data
Commodities A B C D
1992 Base Price 3 5 4 1
Quantity 18 6 20 14
1993 Price 4 5 6 3
Current Quantity 15 9 26 15
Year

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18.36 STATISTICS

The Passche price Index number is :

(a) 146.41 (b) 148.25 (c) 144.25 (d) None of these.
81. From the following data
Commodity Base Year Current Year
Price Quantity Price Quantity
A 7 17 13 25
B 6 23 7 25
C 11 14 13 15
D 4 10 8 8
The Marshall Edge Worth Index number is :
(a) 148.25 (b) 144.19 (c) 147.25 (d) 143.78
82. The circular Test is an extension of
(a) The time reversal Test. (b) The factor reversal Test.
(c) The unit Test. (d) None of these.
83. Circular test, an index constructed for the year ‘x’ on the base year ‘y’ and for the year ‘y’ on
the base year ‘z’ should yield the same result as an index constructed for ‘x’ on base year ‘z’
i.e. I01 × I12 × I20 equal is :
(a) 3 (b) 2 (c) 1 (d) None of these.
84. In 1976 the average price of a commodity was 20% more than that in 1975 but 20% less than
that in 1974 and more over it was 50% more than that in 1977. The price relatives using 1975
as base year (1975 price relative = 100) then the reduce data is :
(a) 8,.75 (b) 150,80 (c) 75,125 (d) None of these.
85. Time Reversal Test is represented by symbolically is :
(a) P01 x Q01 = 1 (b) I01 x I10 = 1
(b) I01 x I12 x I23 x …. I(n–1)n x In0 = 1 (d) None of these.
86. The prices of a commodity in the years 1975 and 1980 were 25 and 30 respectively, taking
1975 as base year the price relative is :
(a) 120 (b) 135 (c) 122 (d) None of these.
87. From the following data
Year 1992 1993 1995 1996 1997
Link Index 100 103 105 112 108
(Base 1992 = 100) for the years 1993–97. The construction of chain index is :
(a) 103, 100.94, 107, 118.72 (b) 103, 108.15, 121.3, 130.82
(c) 107, 100.25, 104, 118.72 (d) None of these.

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 INDEX NUMBERS 18.37

88. During a certain period the cost of living index number goes up from 110 to 200 and the
salary of a worker is also raised from ` 330 to ` 500. The worker does not get really gain.
Then the real wages decreased by :
(a) ` 45.45 (b) ` 43.25 (c) ` 100 (d) None of these.
89. Net monthly salary of an employee was ` 3000 in 1980. The consumer price index number in
1985 is 250 with 1980 as base year. If the has to be rightly compensated then, 7 th dearness
allowances to be paid to the employee is :
(a) ` 4.800.00 (b) ` 4,700.00 (c) ` 4,500.0 (d) None of these.
90. Net Monthly income of an employee was ` 800 in 1980. The consumer price Index number
was 160 in 1980. It is rises to 200 in 1984. If he has to be rightly compensated. The additional
dearness allowance to be paid to the employee is :
(a) ` 240 (b) ` 275 (c) ` 250 (d) None of these.
91. When the cost of Tobacco was increased by 50%, a certain hardened smoker, who maintained
his formal scale of consumption, said that the rise had increased his cost of living by 5%.
Before the change in price, the percentage of his cost of living was due to buying Tobacco is
(a) 15% (b) 8% (c) 10% (d) None of these.
92. If the price index for the year, say 1960 be 110.3 and the price index for the year, say 1950 be
98.4, then the purchasing power of money (Rupees) of 1950 in 1960 is
(a) ` 1.12 (b) ` 1.25 (c) ` 1.37 (d) None of these.

93. If P Q
0 0 = 1360 ,  PnQ 0 = 1900 , P Q
0 n = 1344 , P Qn n = 1880 then the Laspeyre’s Index
number is
(a) 0.71 (b) 1.39 (c) 1.75 (d) None of these.
94. The consumer price Index for April 1985 was 125. The food price index was 120 and other
items index was 135. The percentage of the total weight index given to food is
(a) 66.67 (b) 68.28 (c) 90.25 (d) None of these.
95. The total value of retained imports into India in 1960 was ` 71.5 million per month. The
corresponding total for 1967 was ` 87.6 million per month. The index of volume of retained
imports in 1967 composed with 1960 (= 100) was 62.0. The price index for retained inputs for
1967 our 1960 as base is
(a) 198.61 (b) 197.61 (c) 198.25 (d) None of these.
96. During the certain period the C.L.I. goes up from 110 to 200 and the Salary of a worker is
also raised from 330 to 500, then the real terms is
(a) Loss by ` 50 (b) Loss by 75 (c) Loss by ` 90 (d) None of these.
[Hint : Real Wage = (Actual wage/Cost of Living Index) * 100]

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18.38 STATISTICS

97. From the following data

Commodities Q0 P0 Q1 P1
A 2 2 6 18
B 5 5 2 2
C 7 7 4 24
Then the fisher’s quantity index number is
(a) 87.34 (b) 85.24 (c) 87.25 (d) 78.93
98. From the following data
Commodities Base year Current year
A 25 55
B 30 45
Then index numbers from G. M. Method is :
(a) 181.66 (b) 185.25 (c) 181.75 (d) None of these.
99. Using the following data
Commodity Base Year Current Year
Price Quantity Price Quantity
X 4 10 6 15
Y 6 15 4 20
Z 8 5 10 4
the Paasche’s formula for index is :
(a) 125.38 (b) 147.25 (c) 129.8 (d) 99.06
100. Group index number is represented by
Price Relative for the year (Pr ice Re lative  w )
(a) ×100 (b)
Price Relative for the previous year w

(Pr ice Re lative  w )

(c)   100 (d) None of these.
w

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 INDEX NUMBERS 18.39

1. (a) 2. (c) 3. (c) 4. (a) 5. (d)

6. (d) 7. (d) 8. (b) 9. (c) 10. (c)
11. (c) 12. (c) 13. (a) 14. (b) 15. (c)
16. (b) 17. (c) 18. (c) 19. (a) 20. (b)
21. (a) 22. (d) 23. (b) 24. (a) 25. (b)
26. (b) 27. (d) 28. (c) 29. (d) 30. (c)
31. (b) 32. (d) 33. (a) 34. (a) 35. (b)
36. (a) 37. (c) 38. (a) 39. (c) 40. (b)
41. (a) 42. (c) 43. (a) 44. (a) 45. (b)
46. (a) 47. (c) 48. (c) 49. (c) 50. (a)
51. (a) 52. (b) 53. (b) 54. (d) 55. (d)
56. (a) 57. (c) 58. (b) 59. (a) 60. (b)
61. (c) 62. (a) 63. (b) 64. (b) 65. (a)
66. (d) 67. (a) 68. (a) 69. (b) 70. (b)
71. (b) 72. (a) 73. (b) 74. (a) 75. (d)
76. (b) 77. (d) 78. (b) 79. (a) 80. (a)
81. (b) 82. (a) 83. (c) 84. (b) 85. (b)
86. (a) 87. (b) 88. (c) 89. (c) 90. (d)
91. (c) 92. (a) 93. (b) 94. (a) 95. (b)
96. (a) 97. (d) 98. (a) 99. (d) 100. (b)

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© The Institute of Chartered Accountants of India