Statistics - Index Number

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Statistics: Index Number

Conceptualization By:

Soumen Roy, B.Com (H), AICWA.



Learning Objectives

Acquaintance with Key Terms

Conceptualizations By: Soumen Roy 2

Introduction to overall concept

Solving of basic problems



Key Terms - Slide I of III

Index NumberPrice Index

* Whole Price Index

eta r ce n exQuantity Index

Value Index

Base Period

Current Period

Conceptualization By Soumen Roy 3



Key Terms - Slide II of III

Simple Aggregate Index NumberSimple Average Price Relative Index

ei hted A re te Index mber

4

* Laspeyre’s Method

* Paasche’s Method

* Fisher’s Ideal Method

* Bowley’s Method

* Marshall-Edgeworth Method

* Kelly’s Method



Key Terms - Slide III of III

Quantity / Volume Index NumberTest of Consistency

* Unit Test

* Time Reversal Test

* Factor Reversal Test

Consumer Price Index Number




Index Number

What is Index Number?….is a statisticalmeasure designed to show changes in variable or a group of related

variables with respect to time, geographic location or other

characteristic.

- For example, if we want to compare the price level of 2009 with what itwas in 2008, we shall have to consider a group of variables such as

price of wheat, rice, vegetables, cloth, house rent etc.,

- We want one figure to indicate the changes of different commodities as

a whole. This is called an Index number.- In general, index numbers are used to measure changes over time in

magnitude which are not capable of direct measurement.




Characteristics of Index Number

Index numbers are specified averages

Index n mbers re ex ressed in ercent e

Index numbers measure changes not

capable of direct measurement.

Index numbers are for comparison.




Uses of Index Numbers

They measure the relative change.They are of better comparison.

The re economic b rometers.

They compare the standard of living.

They provide guidelines to policy.

They measure the purchasing power of money.




Types of Index Numbers

Price Index: Compares the prices for a group of commodities ata certain time as at a place with prices of a base period. The wholesale

price index reveals the changes into general price level of a country,

but the retail price index reveals the changes in the retail price of

, , .

Quantity Index: Is the changes in the volume of goods

produced or consumed. They are useful and helpful to study the output

in an economy.

Value Index: Compare the total value of a certain period with

total value in the base period. Here total value is equal to the price of

commodity multiplied by the quantity consumed.




Notations

The following notations would be usedthrough out the presentation:

P1 = Price of current year

P0 = Price of base year

q1 = Quantity of current year

q0 = Quantity of base year




Problems in construction of Index Numbers

Purpose of the index numbersSelection of base period

election of items

Selection of source of data

Collection of data

Selection of average

System of weighting




Method of construction of IndexNumbers:

UnWeighted Weighted


Aggregate

Index

Numbers

Simple

Average

of Price

Relative

Aggregate

Index

Number

Weighted

Average

of Price

Relative



Simple Aggregate Index Number

The price of the different commodities of the current year are added

and the sum is divided by the sum of the prices of those commoditiesby 100. Symbolically:

Simple aggregate price index = P01 = ∑P1 / ∑P0 * 100

simple aggregate method taking prices of 2000 as base.

13

Commodity Price Per Unit (In Rupees)

Year: 2000 Year: 2004

A 80 95

B 50 60

C 90 100

D 30 45



Simple Aggregate Index Number

Solution 1:

Commodity Price Per Unit (In Rupees)

Year: 2000 (P0) Year: 2004 (P1)

A 80 95

Simple aggregate price index = P01 = ∑P1 / ∑P0 * 100

= 300 / 250 * 100 = 120.

14

B 50 60C 90 100

D 30 45

Total 250 300



Simple Average Price RelativeIndex First calculate the price relative for the various commodities and then

average of these relative is obtained by using arithmetic mean andgeometric mean.

P01 = [∑ P1 / P0 *100] / n, where n is the number of commodities.

Simple average of price relative by Geometric Mean:

P01 = Antilog [ ∑ log (P1 / P0 *100)] / n

15



Simple Average Price RelativeIndex Example 2: From the following data, construct an index for 2004

taking 2000 as base by the average of price relative using (a)arithmetic mean and (b) Geometric mean.

Commodity Price in 2000 Price in 2004

16

A 50 70

B 40 60

C 80 100

D 20 30



Simple Average Price RelativeIndex Solution: (a) Price relative index number using Arithmetic Mean

Commodity Price in 2000

(P0)

Price in 2004

(P1)

P1 / P0 * 100

A 50 70 140

Simple average of price relative index = (P01) = [∑ P1 / P0 *100] / n

= 565 / 4 = 141.25

17

B 40 60 150

C 80 100 125

D 20 30 150

Total 565



Simple Average Price RelativeIndex Solution: (b) Price relative index number using Geometric Mean

Commodi

ty

Price in

2000 (P0)

Price in

2004 (P1)

P1 / P0 *

100

log(P1/P0

*100)

A 50 70 140 2.1461

Simple average of price relative index = (P01) = Antilog [ ∑ log (P1 /

P0 *100)] / n = Antilog 8.5952 / 4 = Antilog [2.1488] = 140.9

18

B 40 60 150 2.1761

C 80 100 125 2.0969

D 20 30 150 2.1761

Total 8.5952



Weighted Aggregate IndexNumbers In order to attribute appropriate importance to each of the items used in

an aggregate index number some reasonable weights must be used.

There are various methods of assigning weights and consequently a

large number of formulae for constructing index numbers have been

devised of which some of the most important ones are:

1. Laspeyre’ s method2. Paasche’ s method

3. Fisher’ s ideal Method

4. Bowley’ s Method

5. Marshall- Edgeworth method

6. Kelly’ s Method

19



Weighted Aggregate IndexNumbers Laspeyre’ s method: The Laspeyre’s price index is a weighted

aggregate price index, where the weights are determined by quantitiesin the base period and is given by:

P01 L = [∑P1q0 / ∑P0q0 ] *100

’ ’

aggregate price index in which the weight are determined by thequantities in the current year. This is given by:

P01 P = [∑P1q1 / ∑P0q1 ] *100

Fisher’ s ideal Method: Fisher’ s Price index number is the geometric

mean of the Laspeyres and Paasche indices Symbolically:P01 F = √[ P01L * P01P]

20



Weighted Aggregate IndexNumbers Fisher’ s ideal Method: It is known as ideal index number because:

(a) It is based on the geometric mean.

(b) It is based on the current year as well as the base year.

(c) It conform certain tests of consistency.

t s ree rom as.

Bowley’ s Method: Bowley’ s price index number is the arithmetic

mean of Laspeyre’ s and Paasche’ s method. Symbolically:

P01 B = [P01L + P01P] / 2

Marshall- Edgeworth method: This method also both the current

year as well as base year prices and quantities are considered.

Symbolically: P01 ME = [ ∑ (q0 + q1) p1 / ∑ (q0 + q1) p0] * 100

21



Weighted Aggregate IndexNumbers Kelly’s Method: The following formula is suggested for constructing

the index number. Symbolically:

P01 K = [∑P1q / ∑P0q ] *100 , where q = (q0 + q1) / 2

Here the average of the quantities of two years is used as weights.

Example 3: Construct price index number from the following data by

app ying (i Laspeyre’s, (ii Paasche’s and (iii Fisher’s Idea

Method.

22

Commodity 2000 2001

Price Qty. Price Qty

A 2 8 4 5B 5 12 6 10

C 4 15 5 12

D 2 18 4 20



Weighted Aggregate IndexNumbers Solution 3:

Commodity p0 q0 p1 q1 p0q0 p0q1 p1q0 p1q1

A 2 8 4 5 16 10 32 20

(i) Laspeyre’ s Price Index = P01 L = [∑P1q0 / ∑P0q0 ] *100= 251 / 172 * 100 = 145.93

23

C 4 15 5 12 60 48 75 60

D 2 18 4 20 36 40 72 80

172 148 251 220



Weighted Aggregate IndexNumbers (ii) Paasche’ s Price Index = P01 P = [∑P1q1 / ∑P0q1 ] *100

= 220 / 148 * 100 = 148.64

(iii) Fisher’s Ideal Index = P01 F = √[ P01L * P01P]

= . .

= √ 21692.49

= 147.28

Interpretation: The results can be interpreted as follows: If 100

rupees were used in the base year to buy the given commodities, we

have to use Rs 145.93 in the current year to buy the same amount of

the commodities as per the Laspeyre’ s formula. Other values give

similar meaning.

24



Weighted Aggregate IndexNumbers Example 4: Calculate a suitable price index from the following data

Commodity Quantity Price

2006 2007

A 20 2 4

Solution 4: Here the as quantities are given in common we can use

Kelly’ s index price number.

25

B 15 5 6C 8 3 2



Weighted Aggregate IndexNumbersCommodity Q p0 p1 p0q p1q

A 20 2 4 40 80

B 15 5 6 75 90

26

Now, P01 K = [∑P1q / ∑P0q ] *100 , where q = (q0 + q1) / 2

i.e., P01 K = 186/139*100 = 133.81

C 8 3 2 24 16

Total 139 186



Weighted Average of PriceRelative Index When the specific weights are given for each commodity, the weighted

index number is calculated by the formula: ∑pw / ∑w, whereW= Weight of the commodity

P = the price relative index

.

Note:

When the base year value P0q0 is taken as weight, i.e., W= P0q0, then

the above becomes Laspeyre’s formula.

When the weights are taken as W=P0q1, then the above becomes

Paasche’s formula

27



Weighted Average of PriceRelative Index Example 5: Compute the Weighted Average index number for the

following data :

Commodity Price Weight

Current Base Year

28

Year

A 5 4 60

B 3 2 50

C 2 1 30



Weighted Average of PriceRelative Index Solution 5:

Commodity P1 P0 W P=P1 / P0 * 100 PW

A 5 4 60 125 7500

B 3 2 50 150 7500

Weighted Average of Price Relative Index = ∑pw / ∑w

= 21000 / 140= 150


C 2 1 30 200 6000140 21000



Quantity / Volume Index Number

The quantity index numbers measure the physical volume of

production, employment and etc. The most common type of thequantity index is that of :

Laspeyre’ s quantity index number = Q01 L = ∑q1p0 / ∑ q0p0

*100

Paasche’s quantity index number = Q01 P = ∑q1p1 / ∑ q0p1 * 100 Fisher’s quantity index number = Q01 F = √ [ Q01 L * Q01 P ]

These formulae represent the quantity index in which quantities of the

different commodities are weighted by their prices.





Example 6: From the following data compute quantity indices by

(i) Laspeyre’ s method, (ii) Paasche’ s method and (iii) Fisher’ s

method.


Commodity Price Total

Value

Price Total

Value

A 10 100 12 180

B 12 240 15 450

C 15 225 17 340




Solution 6: Here instead of quantity, total values are given. Hence first

we find the quantities of base year and current year, i.e.,Quantity = Total Value / Price.

Com. P0 q0 P1 q1 P0q0 P0q1 P1q0 P1q1

(i) Laspeyre’ s quantity index number = Q01 L = ∑q1p0 / ∑ q0p0

*100 = 810 / 565 *100 = 143.36


A 10 10 12 15 100 150 120 180B 12 20 15 30 240 360 300 450

C 15 15 17 20 225 300 255 340

565 810 675 970




(ii) Paasche’s quantity index number = Q01 P = ∑q1p1 / ∑ q0p1 *

100 = 970 / 675 *100 = 143.70. (iii) Fisher’s quantity index number = Q01 F = √ [ Q01 L * Q01 P ]

= √ [ 143.36 * 143.70] = 143.53.




Test of Consistency of IndexNumbers Several formulae have been studied for the construction of index

number. The question arises as to which formula is appropriate to agiven problems. A number of tests been developed and the important

among these are:

(1) Unit test: …. re uires that the formula for constructin an index

should be independent of the units in which prices and quantities arequoted. Except for the simple aggregate index (unweighted) , all other

formulae discussed here satisfy this test.

(2) Time Reversal test: ….the formula for calculating the index

number should be such that it gives the same ratio between one point

of comparison and the other, no matter which of the two is taken as

base.




Test of Consistency of Index

Numbers Symbolically, the following relation should be satisfied.:

P01 * P10 = 1, Where P01 is the index for time ‘ 1’ as time ‘ 0’ asbase and P10 is the index for time ‘ 0’ as time ‘ 1’ as base. If the

product is not unity, there is said to be a time bias is the method.

’

Proof: P01 F = √ [ ∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1]

P10 F = √ [ ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0]

Then P01 F * P10 F = √ [ ∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1* ∑P0q1 /

∑P1q1 * ∑P0q0 / ∑P1q0] = √1=1

Therefore Fisher ideal index satisfies the time reversal test.





Numbers (3) Factor Reversal test: ….holds that the product of a price index

and the quantity index should be equal to the corresponding valueindex. In other word, if P01 represent the changes in price in the

current year and Q01 represent the changes in quantity in the current

year, then P01 *q01 = ∑P1q1 / ∑P0q0.

Fisher’ s ideal index satisfies the factor reversal test.Proof:

P01 F= √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1]

Q01F = √ [∑q1P0 / ∑q0P0 * ∑q1P1 / ∑q0P1]

Then P01 F * q01F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q01* ∑q1P0 / ∑q0P0 * ∑q1P1 / ∑q0P1] = √ [∑P1q1 / ∑P0q0 ]² = ∑P1q1 / ∑P0q0

Therefore Fisher ideal index satisfies the time reversal test.





Numbers Example 7: Construct Fisher’ s ideal index for the following data. Test

whether it satisfies time reversal test and factor reversal test.

Commodity Base Year Current Year

Quantity Price Quantity Price


A 12 10 15 12

B 15 7 20 5

C 5 5 8 9




Numbers Solution 7:

Com q0 P0 q1 P1 P0q0 P0q1 P1q0 P1q1

A 12 10 15 12 120 150 144 180

Fisher’s Ideal Index = P01F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1]

*100

= √[ 264 / 250 * 352 / 330] * 100

= √1.056 * 1.067] *100 = 106.12


C 5 5 8 9 25 40 45 72

250 330 264 352




Numbers Time Reversal Test: This is satisfied when P01 * P10 = 1.

Now, P01 F = √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1]= √ [264 / 250 * 352 / 330]

And P101 F = √ [ ∑P0q1 / ∑P1q1 * ∑P0q0 / ∑P1q0]

=

Then, P01 F * q01F = √ [264 / 250 * 352 / 330 * 330 / 352 * 250 /

264] = √ 1 = 1.

Hence Fisher ideal index satisfy the time reversal test.





Numbers Factor Reversal Test: This is satisfied when P01 *q01 = ∑P1q1 /

∑P0q0.Now, P01 F= √ [∑P1q0 / ∑P0q0 * ∑P1q1 / ∑P0q1]

= √ [264 / 250 * 352 / 330]

=√ [330 / 250 * 352 / 264]

Then, P01 *q01 = √ [264 / 250 * 352 / 330 * 330 / 250 * 352 / 264]

= √ [ (352 / 250)² ] = 352 / 250

= ∑P1q1 / ∑P0q0

Hence Fisher ideal index number satisfy the factor reversal test.




Consumer Price Index

Also called the cost of living index.

It represent the average change over time in the prices paid by the

ultimate consumer of a specified basket of goods and services.

A change in the price level affects the costs of living of different

classes of people differently.

The scope of consumer price is necessary, to specify the population

group covered. For example, working class, poor class, middle class,richer class, etc and the geographical areas must be covered as urban,

rural, town, city etc.




Use of Consumer Price Index

Very useful in wage negotiations, wage contracts and

dearness allowance adjustment in many countries.

At government level, the index numbers are used for wage

policy, price policy, rent control, taxation and general

economic policies.

Change in the purchasing power of money and real can be

measured.

Index numbers are also used for analyzing market price for

particular kinds of goods and services.




Method of Constructing


Methods of Construction of CPI

ggregate xpen tureMethod / AggregateMethod

Family Budget Method / Method of WeightedRelative




Method of Constructing


Aggregate Expenditure method: This method is based upon the

Laspeyre’ s method . It is widely used. The quantities of commoditiesconsumed by a particular group in the base year are the weight. The

formula is Consumer Price Index number = ∑P1q0 / ∑P0q0

Famil Bud et method or Method of Wei hted Relatives: This

method is estimated aggregate expenditure of an average family on

various items and it is weighted. The formula is Consumer Price index

number = ∑Pw / ∑w, Where P = (P1 / P0 * 100) for each item. w =

value weight i.e., P0q0.

Note: “Weighted average price relative method” which we have

studied before and “Family Budget method” are the same for findingout consumer price index.





Example 8: From the following calculate the cost of living index using

Family Budget Method taking 2000 s base year.

Items Weights Price in 2000 (Rs) Price in 2004 (Rs)


Food 35 150 140

Rent 20 75 90

Clothing 10 25 30

Fuel & Lighting 15 50 60

Miscellaneous 20 60 80




Solution (8):

Items W P0 P1 P = P1/P0 *100 PW

Food 35 150 140 93.33 3266.55

Rent 20 75 90 120.00 2400.00

Consumer price index by Family Budget method = ∑Pw / ∑w

= 11333.15 / 100 = 113.33.


Clothing 10 25 30 120.00 1200.00

Fuel & Lighting 15 50 60 120.00 1800.00

Miscellaneous 20 60 80 133.33 2666.60

100 11333.15



Acknowledgements

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