Index Number in Statistical Techniques

RAB 8 Barisal

Introduction

An index number is device, which shows by its variation the changes in a magnitude which is not capable of accurate measurement in it or of direct valuation in practice.

Index numbers occupy a place of great prominence in business statistics. Though originally developed for measuring the effect of change in prices, there is hardly any field today where index numbers are not used. They are used to feel the pulse of economy and they have come to be used as indicators of inflationary or deflationary tendencies. In fact, they are described as barometers of economic activity. For example- if one wants to get an idea as to what is happening to an economy he should look to important indices like the index number of industrial production, agricultural production, business activity etc.

The most familiar indices are consumer price index (CPI); it measures the change in the price of a large group of items consumers purchase. Producer price index (PPI) which measures price fluctuations at all stages of production. Dow Jones Industrial Average (DJIA), this is an index of stock prices, but perhaps it would be better to say it is an “indicator” rather than an index. S&P 500 index, the full name of this index is the standard and Poor’s composite index of stock prices. It is an aggregate price index of 500 common stocks. There are many other indices, such as the NIKKEI 225, NASDAQ etc.

Uses Of Index Numbers

Index numbers are indispensable tools of economy and business analysis. Their significance can be best appreciated by the following points.

(1) They Help In Financing Suitable Policies:

Many of the economic and business policies are guided by index number.

(2) They Reveal Trends And Tendencies:

since the index number are most widely used for measuring change over a period of time the time series so formed enable us to study the general trend of the phenomenon under study.

(3) Index Number Is Very Useful In Deflecting:

Index numbers are used to adjust original data for price change, or to adjust wage for cost living changes and thus transform nominal wages to real wages. Moreover nominal income can be transformed into real income and nominal sales into real sales through index number.

Why Convert Data Into Index

  • Index allows us to express change in price, quantity or values in percent.
  • Helps to determine the inflation rate and overall consumer prices change.
  • Make easier to assess the trend –in a series composed of exceptionally large number.

Simple Index

The relative change in price, quantity or value compared to a base period.

Formula:

Index Number in Statistical Techniques

Math:

Divisions

Population

Index

Dhaka

47895218

318.99

Chittagong

20148759

134.19

Rajshahi

41025698

273.24

Khulna

20132548

134.08

Sylhet

10236987

68.179

Barisal

15014789

100

 

 

 

 

 

Graph:

Index Number in Statistical Techniques

 Method Of Calculating Of Index Numbers:

Index numbers are typically two types. These are as follows:

  1. Unweighted index number
  2. Weighted index number

Unweighted Index:

When in index is not weighted by a special weight then that is called unweighted index. It indicates that there is an equal weight for each item.

 

Types Of Unweighted Index:

Index Number in Statistical Techniques

(1) Simple Average Of Price Relative:

Where individual index of each item is divided by number of item.

 

Formula:

Index Number in Statistical Techniques

Math:

Items

2001 Price (Per Unit)

2006 Price(Per Unit)

Index

Bread

8

12

150

Eggs

20

22

110

Milks

26

38

146.15

Apples

80

120

150

Orange

40

60

150

Coffee

90

150

166.67

264

402

872.82

 

 

 

 

 

 Graph:

Index Number in Statistical Techniques

Findings:

This indicates that the mean of the group of indexes increased by 45.47 percent from 2001 to 2006.

 

Limitation of the Simple Average of Price Indexes:

It fails to consider the relative importance of the items included in the item.

 

Advantage of this method:

We would obtain the same value for the index regardless of the units of measures.

(2) Simple Aggregate Index:

 Here we firstly sum the prices (rather than indexes) for two periods and then determine the index based on the totals.

Formula:

Index Number in Statistical Techniques

 

Math:  

Items

2001 Price (Per Unit)

2006 Price(Per Unit)

Bread

8

12

Eggs

20

22

Milks

26

38

Apples

80

120

Orange

40

60

Coffee

90

150

 Index Number in Statistical Techniques

Math:

Index Number in Statistical Techniques

Index Number in Statistical Techniques

 

Findings:

This indicates that the mean of the group of indexes increased by 52.27 percent from 2001 to 2006.And here, all the items influence equally to find out index.

Limitation of the simple aggregate method:

  1. The units in which prices of commodities are given affect the price index.
  2. No consideration is to the relative importance of the commodities.

Weighted Index

Already we know that we can contract an index number by two ways. The second or last one is weighted indices.

 

Definition of weighted indices:

The process of construction of useful index numbers, where a weight is assigned depending on price and quantity sold.

Types of weighted indices:

a)      weighted average of relative index number

b)      weighted aggregate index number

Note that weighted average of relative index number is not so popular for calculating index number rather than weighted aggregate index number. So we mainly focusing on weighted aggregate index number.

Weighted aggregate index number:

This is same as simple aggregate type. Only the fundamental difference that weight are assigned to the various item included in the index. Many famous people have introduced large number of formulas. Here is some example of important ones.

a)      Laspeyres’ method

b)      Paasche’s  method

c)      Fisher’s ideal model

Laspeyres’ method:

Laspeyre use base year quantities as weight.

Formula:

Index Number in Statistical Techniques

Example:  

Index Number in Statistical Techniques

 Index Number in Statistical Techniques

Index Number in Statistical Techniques

Formula:

Index Number in Statistical Techniques

Index Number in Statistical Techniques

Advantages:

a)      Only the base period quantity data is required

b)      Help the more meaning full comparison over time

c)      The change in index indicate the change in price

Disadvantage:

a)      Does not reflect the change in buying pattern over time

b)       Over weight goods whose price and sold quantity is large

 

Paasche’s method:

Paasche use current year quantities as base.

 

Example:

Items

2001 Price(P0)

2001 Quantity

(q0)

2006 Price(Pt)

2006

Quqntity

(qt)

(P0qt)

(Ptqt)

Bread

8

90

12

98

784

1176

Eggs

20

500

22

674

13480

14828

Milks

26

50

38

60

1560

2280

Apples

80

10

120

15

1200

1800

Orange

40

30

60

35

1400

2100

Coffee

90

5

150

7

630

1050

 

Advantages:

a)      It use current period quantities

b)      It reflect current buying habit

Disadvantage:

a)      It requires quantity data for each year –is difficult to obtain, because different quantities are used each year.

b)      It tend to over weight the goods whose price has declined

c)      It requires the price to be computed each year

Fisher’s ideal model:

we know that Laspeyres and Paasche’s index over weight the goods respectively, whose price have increase and whose price have gone down. To over come this problem Irving Fisher introduced a new method is known as Fisher’s ideal index. This index is the geometric mean of the Laspeyres and Paasche’s index.

 

Formula:  Fisher’s ideal index =

                                                    

 

Example:  Fisher’s ideal index =

                                                    = 

 

                                                    = 

                                                    =       122.23

Fisher’s ideal model:

Advantages:

a)      It balance the effect of Laspeyres and Paasche’s index

 

Disadvantage:

  1. It is rarely use in practical, because it has same basic set of problem identified in Paasche’s method
  2. It requires a new set of quantities be determined each year

Special –purpose indexes:

Index number is more significant in the world. Including business, education research, Labor union, financial institution, prepares indexes of prices, quantity and value that are important to their particular field of interest. Here are few examples of special purpose index.

 

  1. The consumer price index (CPI) or cost of living index.
  2. The producer price index (PPI) or whole sale price index.
  3. Dow Jones industrial average index (WJI).
  4. Standard & poor 500 stock index (S&P 500 index).

The consumer price index (CPI) or cost of living index:

It describes the change in prices from one period to another for “market basket” of goods and services. The CPI is not one index. It may be for different cities, countries. The CPI is also called deflator. Generally laspeyres’ method is used to determine CPI.

History of CPI:

  1. The CPI originated in 1913.
  2. Since 1921 it is publishing regularly.
  3. In January 1978 it starts to publish for two groups by Bureau of labor Statistics.
  4. The index includes about 400 items and 250 agent collect monthly.
  5. The current base period is 1982-84.

Special uses of CPI:

 CPI uses in the following cases. They includes-

  1. For calculating real or deflated income.
  2. Deflating sales
  3. Purchasing power of Dollar
  4. Cost of living adjustment.

]

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Annual

1999

100

2000

100.3

100.9

101.6

101.6

101.7

102.1

102.3

102.3

102.8

102.9

102.8

102.6

102

2001

103.3

103.7

103.9

104.2

104.6

104.8

104.5

104.6

104.9

104.7

104.4

103.9

104.3

2002

104.2

104.5

105.1

105.6

105.6

105.6

105.7

106

106.3

106.4

106.3

106

105.6

2003

106.5

107.3

107.9

107.7

107.5

107.6

107.7

108.2

108.5

108.4

108

107.8

107.8

2004

108.5

109.1

109.7

110

110.6

110.8

110.7

110.7

111

111.6

111.6

111.2

110.5

2005

111.3

111.9

112.6

113.4

113.3

113.2

113.7

114.3

115.6

115.7

114.9

114.4

113.7

2006 115.2(U) 115.5(U) 116.1(U) 116.8(U) 117.3(U) 117.5(U) 117.7(U) 117.9(U) 117.7(U) 117.2(U) 117.0(U) 117.1(U) 116.9(U)
2007 117.427(I) 118.030(I) 118.962(I) 119.552(I) 120.041(I) 120.230(I) 120.157(I) 120.077(I) 120.423(I) 120.700(I)

 

Calculating real or deflated income:

 It indicates the how the buying power of consumer is changing with the change of commodities.

Real income =

 

                                            Example:

year

money income

CPI

Real Income

1999

15000

100

15000

2000

18000

102

17647.06

2001

21000

104

20134.23

2002

21500

106

20359.85

2003

22300

108

20686.46

2004

25000

111

22624.43

2005

25500

114

22427.44

2006

27000

117

23096.66

 

Here we found the real income from year 1999-2006.

 

Deflating sales:

 It indicates the change of sales with change of manufacturing materials.

Deflating sales =

 

                                              Example:

Year

Yearly sales

PPI

Deflated sales

1999

125000

100

125000

2000

132250

102

129656.86

2001

150300

104.3

144103.55

2002

152600

105.6

144507.58

2003

185260

107.8

171855.29

2004

205486

110.5

185960.18

2005

206549

113.7

181661.39

2006

250789

116.9

214532.93

Here we see total sales has increased by taka (250789-125000) 125789 from year 1999-2006, but the actual or deflated sales has increased by taka (214532.93-125000) 89532.93. That means by using PPI a producer can get his/her actual sales increases or decreases over the time period.

 

Purchasing power of Dollar:

For the cause of inflation every year the value of money decreases and to find out the purchasing power of money we use an index called “purchasing power of money index” which is described below:

 

It determines what happens purchasing of dollar.

Purchasing power of Dollar=

 

Example:

year

Value Of dollar

CPI

Purchasing Power Of Money

1999

1

100

1

2000

1

102

0.98

2001

1

104.3

0.96

2002

1

105.6

0.95

2003

1

107.8

0.93

2004

1

110.5

0.9

2005

1

113.7

0.88

2006

1

116.9

0.86

 

 

Cost of living adjustment:

CPI is used to adjust wages, pension and so on. For example-a person receives 1000 taka as pension and if the CPI change by 10 points (from 70 to 80) and each point increase by 2%. Then his pension will be 1000(10) (.02) =1200.

Other uses of CPI:

  1. Index number is used for analysis markets for particular kind goods and services.
  2. It is used to measures to changing purchasing power of the currency, real income etc.
  3. At government level it used for wage policy, price policy, rent con troll, taxation and general economics policy.

Limitation of index number:

  1. It is not possible to take each and every item in account. Normally it based on simple.
  2. It is random sampling but it could neither practical nor representative.
  3. It is difficult take account changes in quantity of product.
  4. Fro larger number of methods it is impossible to present right one.

Shifting the base:

Consumption busket of a family is not same from time to time and the expenditures for it also not same. If two or more time series have the same base period, they can be compared directly. A problem arises when two or more series being compared do not have the same base period. When there is no sure the base periods are the same, then a direct comparison is not appropriate.

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