Teach GCSE MathsIndex Index NumbersNumbers

Ind

ex

Farmland Birds

Woodland Birds

U.K. Bird Populations

Year

1970 19901980

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

© Christine Crisp

Index NumbersIndex NumbersIndex NumbersIndex Numbers

Data from the Office for National Statistics which is included in this presentation is reproduced under the terms of the click-use licence.

Many things increase in price from one year to another.The following table shows the prices of a toy and a game in 2000 and 2005.

We can easily see that the price of the toy has increased more than that of the game.

2000 2005

Price of toy (£) 10 20

Price of game (£) 10 15

2000 2005Increas

e

Price of toy (£) 10 20 10

Price of game (£) 10 15 5

Toy

Game

Year

Pri

ce

(£)

Now let’s compare the prices of the same toy with a different game.

The toy has increased by the same amount as the 2nd game . . .

or has it ?

2000 2005

Price of toy (£) 10 20

Price of 2nd game (£)

20 30

2000 2005Increas

e

Price of toy (£) 10 20 10

Price of 2nd game (£)

20 30 10

There are different ways of comparing price increases.

Toy

2nd Game

YearP

rice

(£)

2000 2005Increas

e

Price of toy (£) 10

Price of 2nd game (£)

10Let’s work out the 2005 prices as a percentage of the prices in 2000.

10 100 = Toy: 20 100 =Gam

e:200% 150%

The price of the toy has increased by 100% since 2000 but the game is only 50% more than its price in 2000.

20 30

We have based this calculation on taking the 2000 prices as 100%.

If we drop the % sign we have an index.

10 20

20 30

Year

2000 2005

Price (£)

IndexPrice

(£)Index

Toy 10 100 20 200

Game 20 100 30 150We now have the following graphs:

Toy2nd Game

Price (£)

Year

Pri

ce

(£)

Toy

2nd Game

Index Numbers

Year

Ind

ex

e.g. The graph shows the price index for the average cost of a house bought by a first time buyer.

Source: Office for National Statistics, Social Trends 37

YearP

rice I

nd

ex

(a) When was the price twice its 1995 value?

In 1995, the index was 100 and the cost was

thousand.£46·5

(b) The index was 304 in 2005. What was the cost?

e.g. The graph shows the price index for the average cost of a house bought by a first time buyer.

Source: Office for National Statistics, Social Trends 37

YearP

rice I

nd

ex

(a) When was the price twice its 1995 value?

Solution:

For twice the price we need 200%.

£46·5

The price had doubled during 2001.

In 1995, the index was 100 and the cost was

thousand.

e.g. The graph shows the price index for the average cost of a house bought by a first time buyer.

YearP

rice I

nd

ex

Solution:

so, in 2005 the cost was

46·5

100304 = £141·36

thousand

304

£46·5

(b) The index was 304 in 2005. What was the cost?

In 1995 the cost was

£46·5

In 1995, the index was 100 and the cost was

thousand.

We can use index numbers to compare changes in quantities other than money.

The next graph shows indexes for the UK populations of woodland and farmland wild birds.

e.g.

Source: Office for National Statistics, Social Trends 34.

Year

Ind

ex

Farmland Birds

Woodland Birds

U.K. Bird Populations

(a)Which year is the base year? (b)Do we know how many birds there were in the base year?

Ans: 1970

Ans: No. Index numbers never give actual values.

e.g.

(c) The index for the farmland birds was(i) 116 in 1977, and (ii) 64 in 1998.

What do you think these numbers mean?Ans: (i) In 1977 there were 16% more

farmland birds than in 1970.

Year

Ind

ex

Farmland Birds

Woodland Birds

U.K. Bird Populations

(ii) In 1998 the population had dropped to 64% of the 1970 value.

116

64

x

x

Percentage changes are obvious from the index values.

SUMMARY

An index number is used to compare changes in quantities over time.

The index for the quantities being compared is usually set to 100 in the base year. The base year is always at the start of the period so all the quantities start at the same point.

e.g. If the index is 100 in the base year, Increase of 20% from base

yearIndex 120:

Decrease to 70% of base year value.Index 70:

Exercise

1. The number of acts of vandalism had increased by 125% by 1995.

Vandalism

Vehicle Thefts

Crime in England and Wales

Source: Office for National Statistics, Social Trends 37

For each statement below choose one of the following:(a)

true(b) false(c) we can’t

tell

There were 1·8 million vehicle thefts in 1981.

2. There were more vehicle thefts than acts of vandalism throughout most of the period shown.3. There were about 4·4 million vehicle thefts in 1995.

Exercise

1. The number of acts of vandalism had increased by 125% by 1995.

Vandalism

Vehicle Thefts

Crime in England and Wales

Source: Office for National Statistics, Social Trends 37

Ans: False

The number had increased to 125% of the 1981 value. This is an increase of 25%.

Solution:

Exercise

Vandalism

Vehicle Thefts

Crime in England and Wales

Source: Office for National Statistics, Social Trends 37

2. There were more vehicle thefts than acts of vandalism throughout most of the period shown.

Ans: We can’t tell

We are given no information about the number of acts of vandalism, only the changes in the number.

Solution:

Exercise

Vandalism

Vehicle Thefts

Crime in England and Wales

Source: Office for National Statistics, Social Trends 37

3. There were about 4·4 million vehicle thefts in 1995. Ans: True

We were told there were 1·8 million in 1981. By 1995, the index has increased to about 244.1·8244100 = 4·4

million

Solution: