The document discusses index numbers, which are used as economic indicators to measure changes in economic activities such as prices, sales, exports, and production over time or between locations. There are two main types of index numbers: fixed base index numbers and chain base index numbers. Fixed base index numbers express current data as a percentage of a fixed base year, while chain base index numbers link indices over successive periods. The document provides an example to illustrate how to calculate a fixed base price index and discusses issues statisticians face when constructing index numbers such as choosing items, base years, and weighting formulas.

1. Chapter#7
Index Numbers
Lecture#10,Prepared from Tools of Statistical Analysis, S. M. Ahsan
Hussain, 12th Edition, Kifayat Academy Educational Publishers,2009.

2. Index Numbers
• Index Number is the most commonly employed tool of Statistics in
the field of Business and Economics. These are considered to be the
economic indicators or economic barometer. These are pure numbers
expressed as percentages of selected base. Index numbers provide a
measure of changes occurring in any economic activity, such as prices,
sales, exports, production etc. from time to time or place to place.
• Two types of index numbers are usually constricted according to the
choice of base.
• Fixed Base index numbers
• Chain base index numbers

3. • The numbers are expressed as percentage of some fixed year as the
base, and comparison is made between the current year and the
fixed base year.
• For example the average price of cotton were Rs. 25, 30, 31, 35, and
40 per Kg during 1995, till 1999, now consider the price of cotton in
1995 to be 100, then the price of cotton 1996 through 1999 would be
120, 124, 140, ands 160. HOW???
Class Activity

4. Year Price(Rs) Index Number
1995 25 100
1996 30 120
1997 31 124
1998 35 140
1999 40 160
Base Year
20%
increase
60%
increase
Fixed Base index numbers

5. 0
20
40
60
80
100
120
140
160
180
1995 1996 1997 1998 1999
Chart Title
Price(Rs) Index Number
Year Price(Rs) Index Number
1995 25 100
1996 30 120
1997 31 124
1998 35 140
1999 40 160

6. Chain base index numbers
Year Price(Rs) Index Number
1995 25 100
1996 30 120
1997 31 103
1998 35 113
1999 40 114
20% increase
in 1996 over
1995
3%
increase
Chain base index numbers have less practical utility than Fixed Base index numbers

7. • https://youtu.be/ZjOjJNG8nmU
• https://sabaq.pk/video-page.php?sid=federal-statistics-11th-
4.1&v=stat-11-12-index-no2&s=st

8. • In the previous example, only a
list of prices in different years is
considered but in actual practice
to know the cost of living, for
example, it is not enough to
know the prices only, but also to
know the amount of different
commodities consumed. These
amounts of different
commodities are known as
weights. Different weighing
procedures are in use with their
own merits over the others.
Year Price(Rs) Index Number
1995 25 100
1996 30 120
1997 31 103
1998 35 113
1999 40 114

9. A hypothetical family consumes only four commodities in the year 2000, and
also the same amounts of these commodities are assumed to be consumed
during 2001. The average price and the quantities consumed during the two
periods are shown in table 7.1, where 𝑃0 𝑎𝑛𝑑 𝑄0 are the price and quantity
respectively for the base year, and 𝑃𝑛 𝑎𝑛𝑑 𝑄𝑛 are the corresponding values the
current year or the years for which the index to be constructed.
2001
Commodities
A 10 25 12 250 300
B 12 25 15 300 375
C 15 14 20 210 280
D 8 20 10 160 200
920 1155
2000
𝑃0 𝑄0 𝑃𝑛 𝑃0 𝑄0 𝑃𝑛 𝑄0
table 7.1
Consumption pattern remain same for the two periods

10. 2001
A 10 25 12 250 300
B 12 25 15 300 375
C 15 14 20 210 280
D 8 20 10 160 200
920 1155
2000
𝑃0 𝑄0 𝑃𝑛 𝑃0 𝑄0 𝑃𝑛 𝑄0
Price and Quantity for the base year 2000
Price for the year for which index is to be constructed
Computations
Total
expenditure
in 2000
1155
920
𝑥 100 = 125.5 i.e. rise of about 25.5% over the base year
Total
expenditure
in 2001

11. A hypothetical family consumes only four commodities. The average price and
the quantities consumed during the two periods are shown in table 7.2, where
𝑃0 𝑎𝑛𝑑 𝑄0 are the price and quantity respectively for the year 2000, and
𝑃𝑛 𝑎𝑛𝑑 𝑄𝑛 are the price and quantity for the year 2001.
Compare the expenditures(Cost of Living) during the two periods(2000 &
2001), by considering quantities of the current year as weight.
Class Activity
10 25 12 30
12 25 15 25
15 14 20 20
8 20 10 25
2000 2001
𝑃0 𝑄0 𝑃𝑛 𝑄𝑛

12. 10 25 12 30 300 360
12 25 15 25 300 375
15 14 20 20 300 400
8 20 10 25 200 250
1100 1385
2000 2001
𝑃0 𝑄0 𝑃𝑛 𝑃0 𝑄𝑛 𝑃𝑛 𝑄𝑛
𝑄𝑛
Expenditure during current 2000 is 1100, and expenditure during 2001 is 1385,
now if expenditure during 2000 is assumed to be 100, the expenditure during
2001 would be
1385
1100
× 100 = 125.9
i.e. rise of about 26% in the cost of living over the previous year.

13. Which quantities were taken as weights in the above
two examples?
Class Activity

14. Food for thought
• In the above tow examples, the quantities of either base year OR the
current year were considered as weights, but it may be argued that
why the average of the base year and the current year may not be
considered as weights?
• Whatever, may be the weighing procedure, the weighted index
numbers give the real change in the level of some phenomenon over
time.

15. Problems faced by
statistician during
Construction of
Index Numbers

16. 1. Specification of Purpose
• Who is going to use these indices?
• The specification of the purpose will help in choice of weights, choice
of items, choice of base year etc.
• Most published indices have not been constructed with any specified
purpose in mind, these are sometimes referred to as general
purpose indices, hence they provide the widest possible use.

17. 2. Choice of items to be included
• It is not possible to include all the commodities within the field of enquiry.
A very large number of items may result in a higher cost of construction
and delay, while smaller number effect accuracy.
• For example while constructing a cost of living index, we require not only
the retail prices of goods, but also the rent, the expenditure on clothing,
education, gas, and electricity etc. These quotations will be different for
different class of persons for whom cost of living index is being
constructed.
The data should be the true representation of the taste and
habits of persons for whom the index is being constructed.

18. 3. Choice of Base
• Irrespective of the formula and the weighing procedures used, it is
customary to select some period as a base with which other indices
are compared.
• A Normal Year is usually considered as The base period .
• The prices, production, exports etc. are always advancing with time,
and therefore no year is sufficiently normal to be a perfect base. It is
therefore, an average of several years considered as base year.
• For purposes of exercise and practice, the year in the beginning of the
period is usually considered as base. The prices and quantities for
base year are denoted by 𝑃0 𝑎𝑛𝑑 𝑄0, whereas for current year by
𝑃𝑛 𝑎𝑛𝑑 𝑄𝑛

19. 4. Choice of the formula and the Weighting
• About 200 different formula are available for the construction of
index numbers, most of them by R. A. Fisher.
• Here, we will consider following formulae:
• Relatives(Simple indices)
• Simple aggregative index numbers
• Marshal Edgeworth formula
• Fisher ideal index number

20. Relatives
The formulae are as follows:
𝐼 =
𝑃𝑛
𝑃0
× 100 (Price Index OR Price Relative)
𝐼 =
𝑄𝑛
𝑄0
× 100 (Quantity Index OR Quantity Relative)
• These are the most crude methods in which the price change or quantity change
of a Single commodity can be observed.
• In order to find the average change in prices, or quantities over several years,
some average of the relatives is computed. It may be the Arithmetic Mean of the
relatives, median of the relatives, OR the Geometric Mean of the relatives.
Usually the geometric mean or median of relatives is computed.

21. Calculate the simple price indices(relatives) from the following
Data with 1995 as base. Also compute the Geometric and
Arithmetic means of the relatives.
Years Price Log I
1995 25 100 2.0000
1996 26 104 2.0170
1997 30 120 2.0792
1998 33 132 2.1206
1999 35 140 2.1461
2000 38 152 2.1818
SUM 748 12.5448
2.0908
𝐼 =
𝑃
𝑛
𝑃0
× 100
. =
𝐼
. = 𝑛 2.090 = 12 .2
On the average, there is about 23 % rise in the prices over the
period 1995-200
𝑟 ℎ𝑚𝑎 𝑐 𝑒𝑎𝑛 𝑓 𝑅𝑒 𝑎 𝑣𝑒𝑠
=
𝐼
=
74
6
= 124.66

22. Simple Aggregate Index Numbers
The relatives or the average of relatives, as previously stated are crude
methods of comparing changes in price OR quantity of a simple
commodity at different dates. If instead, there are a group of
commodities and the overall change of the prices or the
quantity(measured in the same units) of the group is to be observed for
a specified period, the relatives or the average of relatives does not
help. In this situation simple modified formulae are used, which are
known as simple aggregate index numbers.
𝐼𝑎 =
𝑃𝑛
𝑃0
× 100 𝐼𝑎 =
𝑄𝑛
𝑄0
× 100
Price Index Quantity Index

23. Calculate the simple aggregative indices of the selected
food crops for the years 2000 and 2001 considering
1999 as base from the following data.
Commodities
1999 2000 2001
Wheat 25 30 40
Rice 30 35 50
Maize 15 17 20
Barley 20 25 28
90 107 138
Price in Rs. Per 2 Kilos
𝐼𝑎(2000) =
𝑃𝑛
𝑃0
× 100
𝟏𝟎𝟕
𝟗𝟎
× 𝟏𝟎𝟎 = 𝟏𝟏𝟖. 𝟗
𝐼𝑎 2001 =
𝟏𝟑𝟖
𝟗𝟎
× 𝟏𝟎𝟎
= 𝟏𝟓𝟑. 𝟑

24. Weighted Average Index Numbers
The two defects pointed out in the explanation of simple aggregative
index numbers may to some extent be removed by the introduction of
of each commodity. These weights are the weights corresponding to
the relative importance amounts of different commodities consumed,
produced, sold, exported, or imported etc., depending on weather on
the cost of living indices, indices of industrial production or wholesale
price indices etc. , are to be computed. Thus, if a cost of living index
number is required, the quantities of different items consumed in a
specified period are considered as weights.

25. LASPEYRE’S INDEX
• The formula for the index numbers with Base Year quantities as
weights was given by LASPEYRE.
𝐼𝑛 =
𝑃𝑛𝑄0
𝑃0𝑄0
× 100
• Generally with the lapse of time, the prices go up.
• Index numbers with base year weighing posses an upward bias
Base Year
quantities are
multiplied with
current year prices
Base year quantities are
multiplied with Base
year Prices

26. Class Activity
Following table shows the prices, and quantities of a few items. Using
quantities of 1999 as weights find weighted aggregative indices for
2000, and 2001 with 199 as base.
Items Units 2000 2001
P Q P P
Food K.G. 25 50 30 32
Clothing Metre 5 10 10 12
Electricitry K.W.H 0.25 100 0.3 0.3
Education - 20 3 25 25
Miscellaneous - 11 5 10 15
1999

27. Items Units 2000 2001 2000 2001
P Q P P
Food K.G. 25 50 30 32 1,250 1,500 1,600
Clothing Metre 5 10 10 12 50 100 120
Electricitry K.W.H 0.25 100 0.3 0.3 25 30 30
Education - 20 3 25 25 60 75 75
Miscellaneous - 11 5 10 15 55 50 75
1,440 1,755 1,900
1999
𝑃0 𝑄0 𝑃𝑛 𝑄0 𝑃𝑛 𝑄0
1,755
1,440
× 100 = 121.
1,900
1,440
× 100 = 1 1.9
Cost of living increases by
about 22% in 2000 over 1999
Cost of living increases by
about 32% in 2001 over 1999

28. Paashe’s Formula
•𝐼𝑛 =
𝑃𝑛𝑄𝑛
𝑃0𝑄𝑛
× 100
•Relative decline in Price is responsible for
increased consumption and vise versa.
•Downward Bias
•This index gives lower limit to change in
price.
Current year
Quantities as
weights

29. Class Activity
Following table contains wholesale prices and quantities sold of four food
grains. Prices per 5KG and the quantities in hundred kilos. Construct indices
for 1996-98 with 1995 as base using some suitable formula.
1995
Item
Wheat 0.75 0.8 325 0.83 315 1.1 300
Rice 1.5 1.75 215 2.25 210 3 200
Maiz 0.5 0.65 40 0.68 45 0.75 50
Gram 0.4 0.45 20 0.5 30 0.6 40
1997 1998
1996
𝑃0 𝑃𝑛 𝑄𝑛 𝑃𝑛 𝑄𝑛 𝑃𝑛 𝑄𝑛

30. 1995
Item
Wheat 0.75 0.8 325 0.83 315 1.1 300 243.8 260.0 236.25 261.45 225 330
Rice 1.5 1.75 215 2.25 210 3 200 322.5 376.3 315 472.5 300 600
Maiz 0.5 0.65 40 0.68 45 0.75 50 20.0 26.0 22.5 30.6 25 37.5
Gram 0.4 0.45 20 0.5 30 0.6 40 8.0 9.0 12 15 16 24
594.25 671.25 585.75 779.55 566.00 991.50
1996 1998
1997
1997 1998
1996
𝑃0 𝑃𝑛 𝑄𝑛 𝑃0 𝑄𝑛 𝑃𝑛 𝑄𝑛
𝑃𝑛 𝑄𝑛 𝑃𝑛 𝑄𝑛 𝑃0 𝑄𝑛 𝑃𝑛 𝑄𝑛 𝑃0 𝑄𝑛 𝑃𝑛 𝑄𝑛
𝑃0 𝑄𝑛 𝑃𝑛 𝑄𝑛
671.25
594.25
× 100 = 112.9
779.55
5 5.75
× 100 = 1 .0
991.5
566
× 100 = 175.1

31. Marshal Edgeworth Formula
• As mentioned, the Laspeyre’s form suffers from an upward bias and the
Paasche’s formula suffers from downward bias. These biases to some
extent can be eliminated by combining the base year and the current year
quantities, and a formula can be designed which do no possesses any
general bias. One such formula was given by Marshal and Edgeworth, and
known as Marshal-Edgeworth formula. This formula uses the average(or
the total) of the base year and the current year quantities as weights.
•𝑰𝒎=
𝑷𝒏
𝑸𝟎+𝑸𝒏
𝟐
𝑷𝟎
𝑸𝟎+𝑸𝒏
𝟐
× 𝟏𝟎𝟎

32. Class Activity
Compute a weighted aggregative index number with
“base year weighing” for 2001 considering 2000 as base,
from the following data.
Year
P Q P Q P Q
2000 9.5 100 8 15 5.5 10
2001 10 150 8 20 6 15
Jawar
Rice Wheat