statistics lecture 12 (chapter 12)

2

• On the economic and business front many concepts can be measured directly

• When it is not possible, need to introduce an associated quantity to represent it

• Referred to as an index number

• Examples of index numbers

– Production Price Index

– Consumer Price Index

– JSE Mining, Industrial, Gold, All Share, Bond indices

– Business Confidence Index

3

• A measure that summarises the change in the level of

activity, price or quantity, of a single item or a basket of

related items from one time period to another

• Expressing the value of an item in the period for which the

index is calculated as a ratio of its value in the base period

• Index is a percentage value

What is an index number?

Period of reference

relative to which an

index is calculated

4

•

• When the value exceeds 100, indicates an increase in the level of activity

• When the value is less than 100, indicates a decrease in the level of activity

• Activity can indicate a change in price or quantity – Price index

– Quantity index

What is an index number? Value in period of interest

Index number = × 100Value in base period

Say: Index = 98.6

There was a:

100 – 98.6 = 1.4%

decrease

Say: Index = 108.4

There was a:

108.4 – 100 = 8.4%

increase

5

• Price indices - P

– Price of the item in the period of interest – pn

– Price of the item in the base period – p0

• Quantity indices - Q

– Quantity of the item in the period of interest – qn

– Quantity of the item in the base period – q0

What is an index number?

6

• Simple price index indicates the change in price of a

single item from the base period to the period under

consideration

Simple index numbers

0

100npP

p

• Simple quantity index indicates the change in quantity of a

single item from the base period to the period under

consideration

0

100nqQ

q

7

• The following table indicates the prices, in rand, and

quantities (in 100) sold at a small supermarket for three years

Simple index numbers - example

2007 2008 2009

Price Quantity Price Quantity Price Quantity

Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

8

• Simple price index for sugar in 2008 with 2007 as base year


0

5.29100 100 106.01 6.01 % increase

4.99

npP

p

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

9

• Simple quantity index for milk in 2009 with 2007 as base year


0

59.2100 100 121.06 21.06 % increase

48.9

nqQ

q

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

10

• Simple quantity index for sugar in 2009 with 2008 as base year


0

18.2100 100 97.3 2.7 % decrease

18.7

nqQ

q

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

Concept Questions

• 1 – 6, p418, textbook

11

12

• Composite index reflect the average change in activity of

a basket of items from the base period to the period

under consideration

– Unweighted composite indices – all items in the

basket is considered to be of the same importance

– Weighted composite indices – each item in the basket

is weighted according to its relative importance

Composite index numbers

13

• Simple composite price index

Unweighted composite index numbers

0

100np

Pp

• Simple composite quantity index

0

100nq

Qq

14

• Simple composite quantity index for 2009 with 2007 as base year

Unweighted composite index numbers - example

0

14.2 18.2 59.2100 100 115.5 15.5 % inc

13.1 17.3 48.9

nqQ

q

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

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• Simple composite price index for 2008 with 2007 as base year

Unweighted composite index numbers - example

0

15.99 5.29 8.99100 100 132.4 32.4 % inc

10.49 4.99 7.39

npP

p

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

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• Weighted composite price index

Weighted composite index numbers

0

100np w

Pp w

• Weighted composite quantity index

0

100nq w

Qq w

Where: w = weight assigned to each item in the basket

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• Weighted composite price index for 2008 with 2007 as base year using the profit for each item as weight

Weighted composite index numbers - example

0

100

15.99(.7) 5.29(.3) 8.99(.2)100 141.3 41.3 % inc

10.49(.7) 4.99(.3) 7.39(.2)

np wP

p w

2007 2008 2009

Price Profit Price Profit Price Profit

Coffee (500g) 10.49 70% 15.99 70% 17.99 70%

Sugar (500g) 4.99 30% 5.29 30% 7.49 30%

Milk (1 l) 7.39 20% 8.99 20% 9.39 20%

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• The base period values will be assigned as weights to

the items in the basket


- Laspeyres approach

• Laspeyres price index

0

0 0

100n

L

p qP

p q

• Laspeyres quantity index

0

0 0

100n

L

q pQ

q p

Price index:

weight is the quantity

in the base period

Quantity index:

weight is the price

in the base period

19

• Advantage is that indices calculated for different period

using the same basket of items may be compared

directly as long as the base period remains unchanged

• Disadvantage is that it over estimates increases in the

prices as times goes by – it is necessary to adjust the

base period from time to time


- Laspeyres approach

20

• Laspeyres price index for 2009 with 2007 as base year


0

0 0

17.99(13.1) 7.49(17.3) 9.39(48.9)100 100

10.49(13.1) 4.99(17.3) 7.39(48.9)

140.9 40.9 % inc

n

L

p qP

p q

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

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• The consumed current period values will be assigned as

weights to the items in the basket


- Paasche approach

• Paasche price index

0

100n n

P

n

p qP

p q

• Paasche quantity index

0

100n n

P

n

q pQ

q p

Price index:

weight is the quantity

in the current period

Quantity index:

weight is the price

in the current period

22

• Advantage is that indices calculated for different period

using the same basket of items may be compared

directly as long as the base period remains unchanged

• Disadvantage is that it over estimates increases in the

prices as times goes by – it is necessary to adjust the

base period from time to time


- Paasche approach

23

• Paasche quantity index for 2009 with 2007 as base year


0

14.2(17.99) 18.2(7.49) 59.2(9.39)100 100

13.1(17.99) 17.3(7.49) 48.9(9.39)

114.9 14.9 % inc

n n

P

n

q pQ

q p

2007 2008 2009


Coffee (500g) 10.49 13.1 15.99 12.8 17.99 14.2

Sugar (500g) 4.99 17.3 5.29 18.7 7.49 18.2

Milk (1 l) 7.39 48.9 8.99 53.6 9.39 59.2

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• May only be used if the indices for Laspeyres and Paasche

have the same base period


- Fischer approach

• Fischer price index

F L PP P P

• Fischer quantity index

F L PQ Q Q

Example The price of bread (rands/bread), meat (rands/kg), Cabbage (rands/cabbage) and wine

(rands/bottle), as well as the quantities (in millions) consumed during 2006, 2007 & 2008

are given in the following table:-

Calculate the:-

1. Simple quantity index for meat in 2008 with 2006 as base year

2. Simple composite price index for 2007, with 2006 as base year

3. Lapeyres price index for 2008 with 2007 as base year

4. Paasche price index for 2008 with 2007 as base year

5. Fischer price index for 2008 with 2007 as base year

6. Simple composite quantity index for 2008 with 2007 as base year

7. Fischer quantity index for 2007 with 2006 as base year

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Price Quantity

2006 2007 2008 2006 2007 2008

Bread 7.0 6.6 8.4 900 1000 900

Meat 44.0 46.0 59.0 600 600 700

Cabbage 7.0 7.3 9.6 5 6 5.5

Wine 30.4 30.4 32.1 90 90 100

EXAMPLE ANSWER

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1. 1) Q = 1000

q

qn = 100

600

700 = 116.67

2) P = 1000

p

pn = 100

4.88

3.90 = 102.15

3) PL = 10000

0

qp

qpn = 100

8.36979

6.46146 = 124.79

4) PP = 1000

n

nn

qp

qp = 100

15.41220

8.52122 = 126.45

5) PF = PL PP = )45.126)(79.124( = 125.62

6) Q = 1000

q

qn = 100

1696

5.1705 = 100.56

7) QL = 10000

0

pq

pqn = 100

35471

36178 = 101.99

QP = 1000

n

nn

pq

pq = 100

5.36312

8.36979 = 101.84

QF = PL QQ = )84.101)(99.101( = 101.92

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The index series

• Collection of indices for the same item or basket of items

constructed for a number of consecutive periods using

the same base period

• The base period will be the period = 100

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The index series - example

• Construct an index series for the monthly electricity

usage for a household – use June as base month

Month April May June July August

Useage (kw) 680 754 820 835 798

82.9 92.0 100 101.8 97.3

0

754100 100 92.0 8 % dec

820

nqQ

q

0

680100 100 82.9 17.1% dec

820

nqQ

q

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Important indices – Consumer price index • Composite price index of a representative basket of consumer goods

and services

• Serves as a measure of relative change in the prices of services and

goods consumed in SA

• Stats SA publish the CPI monthly

• Price information in the index refers to the first 7 days of that month

• Published in the second half of the next month

• Info used to determine the CPI is obtained from a survey in each of

12 urban areas for each of 3 income groups and contains almost 600

items in 17 categories

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Important indices – Consumer price index • A weight is assigned to each item in the basket according to their

relative importance

0 0

0 0

p qw

p q

0

pn

pw

CPIw

31

Important indices – Consumer price index • CPI is used to determine the inflation rate

• Deflate other value series

• Adjust prices, wages, salaries and other variables for changes in the

inflation rate

• It is available quickly

• A disadvantage is that it is based on a household with on average 1.6

children, takes only certain good and services into account, includes

indirect taxes but excluded direct taxes

• Can use consecutive CPI’s as a time series to make forecasts on

future values and trends

EXAMPLES OF IMPORTANT INDICES

• JSE all share index

• JSE gold index

• CPI- consumer price index – used to calculate

inflation rate and cost of living

• Inflation rate

• PPI – Production price index

• Business confidence index

• New car sales index

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Statistics SA

• Look at www.stassa.gov.za

33

http://www.stassa.gov.za/

Example

• Activity 1, p197 Module Manual

34

Example

• Activity 2, p199 Module Manual

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Classwork/Homework

• Revision exercises 1,2,3 p 200 module

manual

36

statistics lecture 12 (chapter 12)

Education