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Index Number
Business Statistics B.Com Notes
CBCS Syllabus notes for all the universities of India 

Index Number and Its Features

Historically index number was started in 1764 to compare the Italian price index in 1750 with price level in 1500. Originally index number was used to measure the effect of changes in prices of various commodities but in today’s world it is used in almost every field. Index number is also used to measure the pulse of the economy as a whole and also called barometer of economic activity.

Index number is simply an indicator of changes in prices and quantities. It is a specialized average designed to measure the change in a group of related variables over a period of time. It offers a device of estimating the relative changes of a variable when measurement of actual changes is not possible.  It is also an indicator of inflationary or deflationary tendencies.

In the words of Croxton and Cowden, “Index number is devices for measuring differences in the magnitude of a group of related variables.”

Following are the various features of index number:

1. Measures of relative changes: Index number measure relative or percentage changes in the variable over time.

2. Quantitative expression: Index numbers offer a precise measurement of the quantitative change in the concerned variable over time.

 3. Specialised Average: Index number show changes in terms of average.

4. Index numbers measures the effect of changes over a period of time.

Uses and limitations of index number

Use of Index Number (why index number is called economic barometer)

Index numbers are highly valuable in business and economics. They provide a good basis for comparison as they are expressed in abstract units of measurement. Some of the Use of Index number is listed below:

1.       Measurement of change in the price level or the value of money: Index number can be used to know the impact of the change in the value of money on different sections of the society.

2.       Knowledge of the change in standard of living: Index number helps to ascertain the living standards of people. Money income may increase but if index number show a decrease in the value if money. Living standard may even decline.

3.       Adjustment in salaries and allowances: Cost of living index number is a useful guide to the government and private enterprises to make necessary adjustment in salaries and allowances of the workers.

4.       Information regarding foreign trade: - Index of exports and imports provides useful information regarding foreign trade.

5.       They are important in forecasting future economic activity: Index number is useful not only in studying the past and present workings of our economy, but they are also important in forecasting future economic activity. Price index numbers serve as a useful guide to the business community in planning and decision making purposes.

Index numbers are called barometers of economic activity because of the above mentioned utilities.

Limitations of index number

Index number suffers from various limitations some of which are listed below:

1. Not completely true: Index number not fully true. The index number simply indicate arithmetical tendency of the temporal changes in the variable.

2. International comparison is not possible: Different countries have different bass of index numbers; these do not help international comparisons.

3. Difference of time: With the passage of time, it is difficult to make comparison of index number with the changing time man’s habits.

4. Limited use: Index numbers are prepared with certain specific objective. If they are used for another purpose they may lead to wrong conclusion.

5. Lack of retail price index number: Most of the index numbers are prepared on the basis of wholesaler prices. But in real life, retail prices are most relevant, but it is difficult to collect retail prices.

ALSO READ: CHAPTER WISE BUSINESS STATISTICS NOTES

Unit 1: Statistical Data and Descriptive Statistics

a. Nature and Classification of data

b. Measures of Central Tendency

c. Measures of Variation

d. Skewness, Moments and Kurtosis

Unit 2: Probability and Probability Distributions

a. Theory of Probability

b. Probability distributions:Binomial-Poisson-Normal

Unit 3: Simple Correlation and Regression Analysis

a. Correlation Analysis

b. Regression Analysis

Unit 4: Index Numbers

Unit 5: Time Series Analysis

UNIT 6: Sampling Concepts, Sampling Distributions and Estimation

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Problems in construction of Index number

Before constructing index numbers a careful thought must be given to the problems in construction of index number which are listed below:

a.       Purpose of index number: There are various types of index number, constructed with different objectives. Before constructing an index number, one must define the objective. Index number is not suitable for all purpose. Every index is of limited and specific use. Failure to decide clearly the purpose of the index would lead to confusion and wastage of time with no fruitful results.

b.       Selection of base year: Selection of base year is another problem in the construction of index number. Base year is the reference year with which comparisons of relative changes are made. It may be a year, a month or a day. The indeed for base period is always taken as 100. The following points should be taken into consideration while selecting  base year:

Ø  The base year should be a normal one.

Ø  The base period should not be too distant in the past.

c.       Selection of goods and services: Every goods and services cannot be included while constructing an index number and hence one has to decide what commodities to be included in constructing index number. The commodities should be selected in such a manner that they are representative of the tastes, habits and behaviour of the people for whom the index is meant.

d.      Selection of price: After the commodities have been selected, the next problem is to obtain price quotations for these commodities. Prices of many commodities vary from place to place and even from shop to shop in the same market. Whether wholesale or retail prices are used is also a problem in construction of index number. Price to be selected must be made of representative places and persons.

e.      Selection of appropriate weights: Weights are very important in constructing index number. The term ‘weight’ refers to the relative importance of the different items in the construction of the index. All items are not of equal importance and hence it is necessary to devise some suitable method whereby the varying importance of the different items is taken into account. This is done by allocating weights. Thus we have broadly two types of indices- unweighted and weighted indices.

f.        Choice of average: Since index numbers are specialised averages a decision has to be made as to which particular average i.e. simple, weighted or geometric average should be used for constructing index numbers.

g.       Selection of appropriate weights: The term weight refers to the relative importance of the different items in the construction of the index. All items are not of equal importance and hence it is necessary to devise some suitable method whereby the varying importance of the different items is taken into account.

h.      Selection of appropriate formula: A large number of formulae have been devised for constructing the index. The problem very often is that of selecting the most appropriate formula.

Different types of index number in Statistics

Index number is of three types: Price index, quantity index and value index.

Price Index: Price index is a measure reflecting the average of the proportionate changes in the prices of a specified set of goods and services between two periods of time. Usually a price index is assigned a value of 100 in some selected base period and the values of the index for other periods are intended to indicate the average percentage change in prices compared with the base period.

Quantity Index: Quantity index is a measure reflecting the average of the proportionate changes in the quantities of a specified set of goods and services between two periods of time. Usually a quantity index is assigned a value of 100 in some selected base period and the values of the index for other periods are intended to indicate the average percentage change in quantities compared with the base period. A quantity index is built up from information on quantities such as the number or total weight of goods or the number of services.

Value Index: Value indeed is a measure reflecting the average of the proportionate changes in the value of a specified set of goods and services between base year and current year. Value of goods and services is obtained by multiplying prices and quantities. Usually a value index is assigned a value of 100 in some selected base period and the values of the index for other periods are intended to indicate the average percentage change in values of compared with the base period.

Methods for calculation of index number

Index numbers are constructed in a number of ways. These are broadly divided under two heads:

i) Unweighted Indices                    ii) Weighted indices

In the unweighted indices weights are not expressly assigned whereas in the weighted indices weights are assigned to the various items. Each of these types may be further divided under two heads:

i) Unweighted indices:

a) Simple aggregative method and

b) Simple average of relatives.

a) Simple Aggregative method: Under this method, Price index will be the aggregate of prices of the given period expressed as a percentage of that of the base period.  Here, I = ∑P_n / ∑P₀ x 100

There are two main limitations of the simple aggregative index:

1. In this type of index, the items with large unit prices will highly influence the index number.

2. No consideration is given to the relative importance of the commodities.

b) Simple Average of Price Relatives: A price relative is ratio of current year price expressed as a percentage with respect to the base year price. Under this method price relative of various items are calculated individually and average of price relative are obtained by dividing sum of price relative with number of items. Here, I = ∑ (P_n / P₀ x 100)/N

Merits of this method:

1. Extreme items do not influence the index as equal importance is given to all of the items.

2. The index is not influenced by the units in which prices are quoted.

Demerits of this method:

1. There is problem of selecting an appropriate average.

2. This method gives equal importance to all the relatives which may not always be correct.

ii) Weighted indices:

a) Weighted aggregative method

b) Weighted average of price relatives

a) Weighted Aggregative method: These indices are of the simple aggregative type with the fundamental difference that weights are assigned to the various items included in the index. Some of the weighted aggregative index number is:

1. Laspeyres Index: It is the most widely used index in practice. In this method the base year quantities are taken as weights. The formula for constructing the index is:

Laspeyres Index number = ∑P_nQ₀/∑P₀Q₀ X 100

Merits of Laspeyres Index Method:

a. It is easy and cheapest method to construct index number.

b. The index number of each year can be compared directly because it considers only base year quantities.

c. Weights are only used for one year i.e., base year.

Demerits of Laspeyres Index Method:

a. It does take into account changes in demand.

b. It tends to overstate prices increases.

c. It does not satisfy time and factor reversal test.

2. Paasche’s Index: This method is not so popular alike Laspeyres index method. In this method the current year quantities are taken as weights. The formula for constructing the index is:

Paasche’s Index number = ∑P_nQ_n/∑P₀Q_n X 100

Merits of Paasche’s Index Method:

a. It takes into account consumption patterns into account because it considers current year’s quantity.

b. It does not overstate prices increases.

Demerits of Paasche’s Index Method:

a. This method is long and expensive as compared to Laspeyres method.

b. This is not a pure index as prices and quantities changes.

c. It does not satisfy time and factor reversal test.

3. Fishers ideal index number: Fishers index is an ideal index number because it considers both current and base year’s prices and quantities. It is the geometric mean of Laspeyres and Paasche’s index and calculated as:

Fisher’s index is regarded as ideal index because:

i)        It considers both base year and current year’s price and quantity.

ii)       It satisfies both time reversal and factor reversal test.

iii)     It is based on Geometric mean which is theoretically considered to be the best average of constructing index number.

iv)     It is free from bias as it considers both current year and base year price and qty.

vii) Prove that fisher’s index satisfies both time reversal test and facto7r reversal test.

b) Weighted average of price relatives: In weighted average of prices relatives, AM or GM is used to find index number. The following steps to be followed:

1. Each items of the period for which the index number is calculated is converted into percentage of the same item in the base period.

2. Multiply the percentages as obtained in step 1 with the weights assigned to each items.

3. Add the results obtained in step 2.

4. Divide the sum obtained in step 3 by the sum of the weights used. Symbolically: I = ∑PV/∑V.

Test of adequacy of Index number formulae:

There are various formulas for construction of index number and the problem is that the selection of most appropriate formula for a given situation. In order to find the most appropriate formula, the following tests are suggested:

a)      Time reversal

b)      Factor reversal test

c)       Unit test

d)      Circular test

a) Time Reversal Test: Time reversal test is a test to determine whether a given period method will work both ways in time, forward and backward. In the words of Fisher, “The test is that the formula for calculating the index number should be such that it will give the same ratio between one point of comparison and the other, no matter which of the two is taken as base.” Only Fisher’s ideal index satisfied time reversal test. Symbolically time reversal test can be written as: P₀₁ * P₁₀ = 1

b) Factor Reversal Test: Factor reversal test holds that the product of a price index and the quantity index should be equal to the corresponding value index. In the words of Fisher, “Just as each formula should permit the interchange of the two items without giving inconsistent results so it ought to permit interchanging prices and quantities without giving inconsistent results, i.e. the two results multiply together should give the true value ratio.“ In other words the change in price multiplied by change in quantity should be equal the total change in value. Only Fisher’s ideal index satisfied time reversal test. Symbolically factor reversal test can be written as: P₀₁ * Q₀₁ =∑P₁Q₁ / ∑P₀Q₀

c) Unit Test: Unit test requires that the formula for construction an index number should be independent of the units in which, or for which, prices and quantities are quoted. This formula is satisfied by all the index number formulas except the simple aggregative index method.

d) Circular Test: This formula is similar to time reversal test method. This test is done where there is a frequent shift in the base on which index number is calculated. If comparison of more than two years is to be made, it is always desirable to shift the original base to the previous year which enables us to adjust the index values from period to period. A test of this shiftability of base is called the circular test. Symbolically circular test can be written as: P₀₁* P₁₂*P₂₀= 1

Cost of living index number (CLI) and its uses

Cost of living index numbers generally represent the average change in prices over a period of time, paid by a consumer for a fixed set of goods and services. It measure the relative changes over time in the cost level require to maintain similar standard of living. Items contributing to consumer price index are generally:

i)           Food

ii)          Clothing

iii)        Fuel and Lighting

iv)        Housing

v)         Miscellaneous.

                Uses of cost of living index:-

i)           CLI numbers are used for adjustment of dearness allowance to maintain the same standard of living.

ii)          It is used in fixing various economic policies.

iii)        Its helps in measuring purchasing power of money.

iv)        Real wages can be obtained with the help of CLI numbers.

Steps in Construction of a Consumer Price Index

The following are the steps in constructing a consumer price index:

1) Decision is to be taken about the class of people for whom the index is useful.

2) Next step is to conduct a family budget enquiry covering the population group for whom the index is to be designed.

3) Obtaining price quotations from the localities in which the class of people concerned reside.

4) After quotations have been collected at average price for each of the items included in the index has to be worked out.

Methods of Constructing the Index

After the above mentioned problems are decided, the following two methods can be used to find cost of living index:

1) Aggregate expenditure method: When this method is applied the quantities of commodities consumed by the particular group in the base year are estimated which constitute the weights. The prices of commodities of various groups for the current year are multiplied with the quantities consumer in the base year. This method is similar to Laspeyres index number and formula for calculation is given below:

Consumer Price Index number = ∑P_nQ₀/∑P₀Q₀ X 100

2) Family budget method: When this method is applied the family budgets of a large number of people for whom the index is meant is carefully studied and the aggregate expenditure of an average family on various items is estimated. These constitute the weights. The weights are thus the value weights obtained by multiplying the prices by quantities consumed. The price relatives of each commodity are obtained and these price relatives are multiplied by the value weight for each item and the product is divided by the sum of weights. Symbolically:

Consumer price index = ∑IW/∑W X 100

Fixed base method and Chain base method

Fixed Base Method: Under this method index number is calculated with a fixed base year. By this method the index number of a given year is not influenced by the variation of prices of any other year.

Uses and limitations of fixed base index number: Same as index number mentioned above

Chain Base Method: Under This method the price of any period or year is related with the price of the immediate previous period or year. It has no relation with the price of fixed based period.

Use of Chain base index number:-         

i)        Direct comparison between two successive periods is possible through link indices.

ii)       This method is useful when weights are changing rapidly.

iii)     It facilitates introduction in new item replacing old one.

iv)     It permits the introduction of new commodities and the detection of old ones without necessitating either the recalculation of entire series or other drastic changes.

Disadvantages of Chain base index number:

i)        This Method involves immense calculation.

ii)       Easy interpretation is lacking.

Difference between chain base method and fixed base method:

CHAIN BASE MEHTOD	FIXED BASED MEHTOD
No fixed base is there.	Base Period is fixed.
Immediately preceding period is taken as base.	Base period is arbitrarily chosen.
Calculation is too long.	Calculation is easy.
During Calculation if there is any error then the Entire calculation is wrong.	This is not so in this method.
If data for any period is missing then subsequent chain indices cannot be computed.	This problem does not arise here.

Meaning of Base Shifting, Splicing and Deflating

Base Shifting

Due to change in business condition and economy, it becomes necessary to change the reference base period of an index number form one period to another without returning to the original row dat. This change of reference base period is called “Shifting the base”. The main reasons for base shifting are:

a) The previous base year is very old and it becomes useless for comparison.

b) It may be desired to compare several index number series which have been computed on different base periods particularly if the several series are to be shown on the same graph.

Splicing

Sometimes index number is available for a longer period of time, and there is substantial revision in base period which results into two indices. So there is a need to convert the two indices into a continuous series. The procedure employed for this conversion is called splicing. The below mentioned formula is used for splicing:

Spliced index no. = (Index No. of current year X Old index of new base year)/100

Deflating

By deflating we mean making allowances for the effect of changing price levels. A rise in price level means a reduction in the purchasing power of money. For example, suppose the prices of rice rises from Rs. 1,500 per quintal in 2010 to Rs. 3,000 per quintal in 2020, it means that in 2020 one can buy only half of rice if he spends the same amount which he was spending in 2010 or in other words, the value of rupee is only 50 paise in 2020 as compared to 2010. Thus the value of a rupee is simply the reciprocal of an appropriate price index written as a proportion. The general expressions may be given thus:

Purchasing power of money = 1\Price index

It shall be clear from above that since the value of money goes down with rising prices the workers or the salaried people are interested not so much in money wages as in real wages i.e., not how much they earn but how much their income or wage will buy. For calculating real wages, we can multiply money wages by a quantity measuring the purchasing power of the rupee or we divide the cash wages by an appropriate price index. This process is referred as deflating. The process of deflating can be expressed in the form of formula as follows:

Real wages = (Money wages/Prices index)*100

Real wage or income index no = Index of money wages/Consumer price index